Commuting vector fields

TL;DR

This paper investigates the structure of the centralizer of a germ of vector fields at the origin in complex space, with a focus on the special case of two-dimensional vector fields.

Contribution

It provides new insights into the properties of the centralizer of vector fields, especially in two dimensions, enhancing understanding of their symmetries and invariants.

Findings

Characterization of the centralizer in dimension two

Identification of special cases with unique properties

Extension of known results to higher dimensions

Abstract

In this paper we study the centralizer $C(X)$ of a germ of vector field at $0\in\C^n$. A particular atention is given to the case of dimension two.

Equations

$$[X,Y]=[X,g.X]=X(g).X+g.[X,X]=0.$$

$$[X,[Y,Z]]=[[X,Y],Z]+[Y,[X,Z]]=0.$$

$$\{z\in ({\mathbb{C}}^n,0)|Y_1(z)\wedge ...\wedge Y_r(z)=0\}$$

$$r(X)=max\{r\in \mathbb{N}|\text{there are g.l.i}{Y}_1,...,Y_r\in C(X)\}.$$

$$1\le r(X)\le min(n,d(X)).$$

$$C(X)\simeq \mathbb{C}\{z_2,...,z_n\}.{\mathbb{C}}^n,$$

$$Y=\sum _{1\le i,j\le n}{a}_{ij}{z}_i\frac{\partial }{\partial z_j},$$

$${\lambda }_i\ne \sum _{1\le j\le n}{k}_j{\lambda }_j,\forall 1\le i\le n,\forall k_1,...,k_n\in {\mathbb{Z}}_{\ge 0}\text{with}\sum _j{k}_j\ge 2$$

$$C(X)={\mathcal{D}}_n\oplus {⟨{\Pi }_{j=1}^n{z}_j^{k_j}.\frac{\partial }{\partial z_i}|{\lambda }_i=\sum _j{k}_j{\lambda }_j⟩}_{\mathbb{C}},$$

$${⟨Y_1,...,Y_s⟩}_{\mathcal{R}}:=\{a_1.Y_1+...+a_s.Y_s\in {\mathcal{X}}_n|a_1,...,a_s\in \mathcal{R}\}$$

$$X(f):=\sum _{i=1}^n{X}_i(z)\frac{\partial f}{\partial z_i}=0.$$

$$\mathcal{M}(X)=\{f\in {\mathcal{M}}_n|X(f)=0\}$$

$$[X,f.X]=X(f).X+f.[X,X]=0.$$

$$C(X)={⟨\frac{\partial }{\partial z_1},...,\frac{\partial }{\partial z_n}⟩}_{\mathcal{I}(X)}.$$

$$\mathcal{D}(X)(z):=\{Y(z)\in {\mathbb{C}}^n|Y\in C(X)\}.$$

$$S:=({\mathbb{C}}^n,0)\setminus V=\{z\in ({\mathbb{C}}^n,0)|di{m}_{\mathbb{C}}(\mathcal{D}(X)(z))<r\}.$$

$$S=\bigcap _{\underset{1\le j\le r}{Y_j\in C(X)}}(Y_1\wedge ...\wedge Y_r=0).$$

$$V:=\{z\in ({\mathbb{C}}^n,0)|Y_1(z)\wedge ...\wedge Y_r(z)\ne 0\}.$$

$$Y(z)=f_1(z).Y_1(z)+...+f_r(z).Y_r(z),$$

$$Y_1\wedge ...\wedge Y_{j-1}\wedge Y\wedge Y_{j+1}\wedge ...\wedge Y_r=f_j.Y_1\wedge ...\wedge Y_r,1\le j\le r,$$

$$\sum _{j=1}^{r}X(f_j).Y_j=[X,Y]=0⟹X(f_j)=0,1\le j\le r.\text{qed}$$

$$=\left\{L|L=\sum _{j=1}^n{\lambda }_j{z}_j\frac{\partial }{\partial z_j}\text{and}\sum _{j=1}^n{\lambda }_j=0\right\}=\{L\in {\mathcal{X}}_n|L\text{is linear and}L(z_1...z_n)=0\}$$

$$\mathcal{A}=\{g.R|g\text{is a homogeneous polynomial of degree k.n}\}.$$

$$d{h}_n\wedge dt=0⟹h_n=h_n(t)⟹k_n=k\forall n\in \mathbb{N}.$$

$$L_{X}Y\wedge X=[X,Y]\wedge X+Y\wedge [X,X]=0⟹$$

$$L_X(f_Y\mu )=(X(f_Y)-\nabla X.f_Y)\mu =0⟹$$

$$\frac{X(f_Y)}{f_Y}=\nabla X.$$

$$\frac{X(f_n/f_1)}{f_n/f_1}=\frac{X(f_n)}{f_n}-\frac{X(f_1)}{f_1}=0⟹\frac{f_n}{f_1}\in \mathcal{M}(X),\forall n\ge 2.$$

$$\omega =i_X\nu =\sum _{j=1}^n(-1{)}^{j+1}{X}_{j}d{z}_1\wedge ...\wedge \widehat{d{z}_j}\wedge ...\wedge d{z}_n,$$

$$Y_1\wedge ...\wedge Y_n:=g.\mu ,$$

Full text

The $1^{st}$ author was supported by CNRS and ANR-16-CE40-0008 project "Foliage".

The $2^{nd}$ author was partially supported by CNPq (Brazil) and University of Rennes 1.

1. 1 Introduction

2. 2 First properties and examples

3. 2.1 General properties.

4. 2.2 More general remarks.

5. 2.3 Examples.

6. 3 The case of dimension two

7. 3.1 The case $r(X)=2$ in dimension two.

8. 3.2 Germs of vector fields with just one irreducible separatrix.

9. 3.3 Generalized curves.

10. 3.4 Homogeneous and quasi-homogeneous vector fields.

11. 3.5 The dicritical case.

1. Introduction

Let $\mathcal{O}_{n}$ be the ring of germs at $0\in\mathbb{C}^{n}$ of holomorphic functions and $\mathcal{X}_{n}$ be the $\mathcal{O}_{n}$-modulus of germs at $0\in\mathbb{C}^{n}$ of holomorphic vector fields. We say that $X,Y\in\mathcal{X}_{n}$ commute if $[X,Y]=0$, where $[.,.]$ denotes the Lie bracket. We denote by $C(X)$ the set of germs of $\mathcal{X}_{n}$ commuting with $X$. Note that $C(X)$ is a $\mathbb{C}$-vector subspace of $\mathcal{X}_{n}$ and its complex dimension will be denoted as $d(X)$. The purpose of this paper is to give examples and properties of the spaces $C(X)$ for a large class of vector fields $X$, in particular in dimension two.

For instance, if $\lambda\in\mathbb{C}$ then $\lambda X\in\mathbb{C}(X)$, so that $d(X)\geq 1$. When $d(X)=1$ then $C(X)=\mathbb{C}.\,X$ and we will say also that $C(X)$ is trivial.

There are examples in which $d(X)=\infty$. For instance, if $X$ has a non-constant holomorphic first integral, say $f$, and $g=\Phi\circ f$, where $\Phi\in\mathcal{O}_{1}$ then $Y=g.\,X\in C(X)$, because $g$ is a first integral of $X$:

[TABLE]

In theorem 1 of § 2.1 we will prove the converse when $n=2$: if $X\in\mathcal{X}_{2}$ and $d(X)=\infty$ then $X$ has a non-constant holomorphic first integral.

Another important observation is that $C(X)$ is a Lie algebra: if $Y,Z\in C(X)$ then $[Y,Z]\in C(X)$, which is a consequence of Jacobi’s identity

[TABLE]

If $X$ has no non-constant meromorphic first integral then $C(X)$ is a finite dimensional Lie algebra (see proposition 4 of § 2.3).

Given $Y_{1},...,Y_{r}\in\mathcal{X}_{n}$, we say that they are generically linearly independent (briefly g.l.i) if $Y_{1}\wedge...\wedge Y_{r}\not\equiv 0$. In other words, the analytic subset of $(\mathbb{C}^{n},0)$ given by

[TABLE]

is proper. Given $X\in\mathcal{X}_{n}$ we define:

[TABLE]

Note that, in general

[TABLE]

Moreover, if $d(X)>r(X)$ then $X$ has a non-constant meromorphic first integral (see proposition 2 in § 2.1).

Let us see some simple examples.

Example 1**.**

Let $X=\frac{\partial}{\partial z_{1}}\in\mathcal{X}_{n}$, $n\geq 2$. Then $Y\in C(X)$ if, and only if, $Y=f(z_{2},...,z_{n}).\,v$, where $f\in\mathcal{O}_{n-1}$ and $v$ is a constant vector field. We can say that

[TABLE]

so that $r(X)=n$ and $d(X)=\infty$.**

Example 2**.**

The radial vector field in $\mathbb{C}^{n}$ is given by $R=\sum_{j=1}^{n}z_{j}\frac{\partial}{\partial z_{j}}$. It is easy to check that $Y\in\mathcal{X}_{n}$ commutes with $R$ if, and only if, $Y$ is a linear vector field

[TABLE]

where $(a_{ij})$ is a $n\times n$ matrix with constant entries. In particular, we have $r(X)=n$ and $d(X)=n^{2}$.**

Example 3**.**

Let $X$ be the diagonal vector field $X=\sum_{j=1}^{n}\lambda_{j}\,z_{j}\,\frac{\partial}{\partial z_{j}}$, where we will assume that $0\neq\lambda_{i}\neq\lambda_{j}$ $\forall$ $i\neq j$. It is easy to see that if $Y\in C(X)$ is a linear vector field then $Y$ is also a diagonal vector field $Y=\sum_{j=1}^{n}\mu_{j}\,z_{j}\,\frac{\partial}{\partial z_{j}}$. If the eigenvalues $\lambda_{1},...,\lambda_{n}$ satisfy the non-resonant conditions:

[TABLE]

then $C(X)=$ the set of diagonal vector fields and $d(X)=r(X)=n$.

However, if the eigenvalues have a resonance, say $\lambda_{i}=\sum_{j=1}^{n}k_{j}\,\lambda_{j}$, then the non-linear vector field $Y=\Pi_{j=1}^{n}z_{j}^{k_{j}}.\,\frac{\partial}{\partial z_{i}}$ commutes with $X$. If we denote the set of diagonal vector fields by $\mathcal{D}_{n}$ then

[TABLE]

where the notation $\left<A\right>_{\mathbb{C}}$ denotes the $\mathbb{C}$-vector space generated by the set $A$. In particular, if the eigenvalues have a resonance then $r(X)=n<d(X)\leq\infty$.**

More examples in the case $n=2$ will be seen in § 2.3. In § 3 the case of dimension two will be studied in detail. We will see that in this case, if $r(X)=2$ then the foliation $\mathcal{F}_{X}$, induced by $X$, is Liouvillian integrable: it can be defined by a closed meromorphic 1-form (see § 3.1).

In § 3.2 we study $C(X)$ when $X\in\mathcal{X}_{2}$ is a generalized curve (see [3]) and has just one separatrix. In this case, we will see that $r(X)=1$ and that, either $d(X)=1$, or $d(X)=\infty$ (if $X$ has a non-constant holomorphic first integral). Recall that a separatrix of $X$ is a germ of curve through the origin, say $\gamma\colon(\mathbb{C},0)\to(\mathbb{C}^{n},0)$, which is regular outside [math] and is $X$-invariant. This means that $\gamma(0)=0$ and $0\neq\gamma^{\,\prime}(t)\in\mathbb{C}.\,X(\gamma(t))$ if $t\neq 0$. In the two dimensional case we say that $X$ is non-dicritical if it has a finite number of irreducible separatrices. Otherwise, we say that it is dicritical.

In § 3.3 we study the so-called generalized curves. We will see that if $X$ is a non-dicritical generalized curve, with an isolated singularity at the origin, $DX(0)$ is nilpotent and has in its reduction of singularities a singularity with non-rational characteristic number then $C(X)$ is trivial.

In § 3.4 we study homogeneous and quasi-homogeneous vector fields. We will see that if $X\in\mathcal{X}_{2}$ is quasi-homogeneous and $r(X)=2$ then $X$ has a non-constant meromorphic first integral. Another result, in this case, is that if $X$ is non-dicritical has an isolated singularity at $0\in\mathbb{C}^{2}$ and $DX(0)$ is nilpotent then $r(X)=1$ and, either $d(X)=1$, or $X$ has a non-constant holomorphic first integral and $d(X)=\infty$ (theorem 5).

In § 3.5 we study the dicritical case, when the vector field $X$ has infinitely many separatrices through $0\in\mathbb{C}^{2}$. We will see that $1\leq d(X)\leq 4$ and we will describe completely the cases $d(X)=3$ and $d(X)=4$: if $d(X)\in\{3,4\}$ then $X$ is linearizable and, modulo a multiplicative constant, $X$ is conjugated to the radial vector field (if $d(X)=4$) or to a linear vector field of the form $z_{1}\frac{\partial}{\partial z_{1}}+n\,z_{2}\frac{\partial}{\partial z_{2}}$, where $n\in\mathbb{N}_{\geq 2}$ (if $d(X)=3$). When $d(X)\leq 2$ we will see that $d(X)=r(X)$ in the dicritical case. We finish this section by fixing more notations that be will used along the paper.

$\widehat{\mathcal{O}}_{n}=$ the ring of formal power le at $0\in\mathbb{C}^{n}$.

$\mathcal{M}_{n}=\mathcal{M}(\mathbb{C}^{n},0)=$ the field of germs at $0\in\mathbb{C}^{n}$ of meromorphic functions.

• 2.

$Diff(\mathbb{C}^{n},0)=$ the group of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ fixing $0\in\mathbb{C}^{n}$.

• 3.

Given a subring $\mathcal{R}\subset\mathcal{O}_{n}$ and vector fields $Y_{1},...,Y_{s}\in\mathcal{X}_{n}$ we denote

[TABLE]

the sub $\mathcal{R}$-modulus generated by the $Y_{j`s}$.

• 4.

Given a germ $Y$ at $0\in\mathbb{C}^{n}$ of holomorphic vector field, function or form, we will denote by $\mu(Y,0)$ its algebraic multiplicity at [math] (the order of the first non-zero jet of $Y$).

• 5.

The ring of holomorphic first integrals of a vector field $X\in\mathcal{X}_{n}$ will be denoted by $\mathcal{I}(X)$: $\mathcal{I}(X)=\{f\in\mathcal{O}_{n}\,|\,X(f)=0\}$.

If $z=(z_{1},...,z_{n})$ is a local coordinate system, and $X=\sum_{i=1}^{n}X_{i}(z)\frac{\partial}{\partial z_{i}}$, then $f\in\mathcal{I}(X)$ iff

[TABLE]

The field of meromorphic first integrals of $X\in\mathcal{X}_{n}$ will be denoted by $\mathcal{M}(X)$:

[TABLE]

Note that $\mathbb{C}\subset\mathcal{I}(X)\subset\mathcal{M}(X)$. A function $f\in\mathcal{M}(X)$ is said to be a pure meromorphic first integral if $f,1/f\notin\mathcal{I}(X)$. Observe that $\mathcal{I}(X).\,X\subset\,C(X)$: if $f\in\mathcal{I}(X)$ then

[TABLE]

For instance, if $X(0)\neq 0$, i.e. $X$ is non-singular, then in some local coordinate system $z=(z_{1},...,z_{n})$ we have $X=\frac{\partial}{\partial z_{1}}$. In this case $\mathcal{I}(X)=\mathbb{C}\{z_{2},...,z_{n}\}$, the ring of convergent power series on the variables $z_{2},...,z_{n}$, and

[TABLE]

Remark 1.1**.**

We have seen that $1\leq r(X)\leq min\{n,d(X)\}$. About this inequality, we would like to remark the following:

• a.

When $r(X)<d(X)$ then $X$ has a non-constant meromorphic first integral.

• b.

When $n=2$, $r(X)<d(X)$ and [math] is an isolated singularity of $X$ then $\mathcal{I}(X)\supsetneq\mathbb{C}$, that is $X$ has a non-constant holomorphic first integral.

• c.

When $r(X)=n$, then $X$ is Liouvillian integrable: there exists a closed meromorphic $(n-1)$-form $\omega$ such that $i_{X}\omega=0$.

Some of the above remarks will be proved in § 2.**

2. First properties and examples

2.1. General properties.

In this section we state some elementary properties that will be used along the paper. Some of the remarks stated in § 1 will be proved here.

Proposition 1**.**

Let $X\in\mathcal{X}_{n}$, $X\not\equiv 0$, with $r(X)=r\geq 1$. Given $z\in(\mathbb{C}^{n},0)$ consider the vector subspace $\mathcal{D}(X)(z)$ of $\mathbb{C}^{n}$ defined as

[TABLE]

Then:

• (a).

The set $V:=\{z\in(\mathbb{C}^{n},0)\,|\,dim_{\mathbb{C}}(\mathcal{D}(X)(z))=r\}$ is the complement of a proper analytic subset on $(\mathbb{C}^{n},0)$. In particular, it is open and dense in $(\mathbb{C}^{n},0)$.

• (b).

$\mathcal{D}(X)$* defines an integrable distribution of dimension $r$ on $V$.*

Proof. For simplicity, from now on we will work with representatives of the germs (of functions or vector fields) without specify their domain: we will write $z\in(\mathbb{C}^{n},0)$ to denote that $z$ belongs to some domain where some representative of the germs are defined.

Consider the set

[TABLE]

Note that $z\in S$ if, and only if, for any $Y_{1},...,Y_{r}\in C(X)$ then $Y_{1}(z)\wedge...\wedge Y_{r}(z)=0$. In particular, we have

[TABLE]

The set $S$ is analytic because it is an intersection of analytic subsets of $(\mathbb{C}^{n},0)$. On the other hand, by the definition of $r=r(X)$ there exist $Y_{1},...,Y_{r}\in C(X)$ such that $Y_{1}\wedge...\wedge Y_{r}\not\equiv 0$. Hence $S$ is a proper analytic subset of $(\mathbb{C}^{n},0)$. This proves (a). Assertion (b) follows from the fact that $C(X)$ is a Lie algebra: $Y,Z\in C(X)$ $\implies$ $[Y,Z]\in C(X)$. ∎

Proposition 2**.**

Let $X\in\mathcal{X}_{n}$, $X\not\equiv 0$, be such that $r=r(X)<d(X)$. Then $\mathcal{M}(X)\neq\mathbb{C}$.

Proof. Let $Y_{1},...,Y_{r}\in C(X)$ be generically independent vector fields of $C(X)$. Since $d(X)>r$ there exists $Y\in C(X)$ such that $Y\notin\left<Y_{1},...,Y_{r}\right>_{\mathbb{C}}$. By the definition of $r=r(X)$ we have $Y\wedge Y_{1}\wedge...\wedge Y_{r}\equiv 0$. Let

[TABLE]

Note that $V$ is the complement of an analytic subset of $(\mathbb{C}^{n},0)$. Moreover, if $z\in V$ then $Y(z)\in\left<Y_{1}(z),...,Y_{r}(z)\right>_{\mathbb{C}}$, so that we can write

[TABLE]

where $f_{1},...,f_{r}\in\mathcal{O}(V)$. Since

[TABLE]

the functions $f_{1},...,f_{r}$ extend to meromorphic functions in a neighborhood of [math]: $f_{j}\in\mathcal{M}_{n}$, $1\leq j\leq r$. Since $Y\notin\left<Y_{1},...,Y_{r}\right>_{\mathbb{C}}$ at least one of the functions is non-constant. Finally, relation (1) implies

[TABLE]

Corollary 2.1**.**

Let $X\in\mathcal{X}_{n}$ having an isolated singularity at $0\in\mathbb{C}^{n}$. Then:

If $r(X)=1<d(X)$ then $\mathcal{I}(X)\neq\mathbb{C}$.

• 2.

If $r(X)<d(X)$ then $\mathcal{M}(X)\neq\mathbb{C}$. In particular, if $n=2$ and $X$ is non-dicritical (has finitely many separatrices through the origin) then $\mathcal{I}(X)\neq\mathbb{C}$.

Proof. If $r=r(X)=1$ and $d(X)>1$, then there exists $Y\in C(X)\setminus\mathbb{C}.\,X$. Note that $Y\wedge X=0$, because $r=1$. Since $X$ has an isolated singularity at [math] we have $Y=f.\,X$, where $f\in\mathcal{O}_{n}\setminus\mathbb{C}$. Finally, $X(f)=0$ and so $f\in\mathcal{I}(X)\setminus\mathbb{C}$. If $r(X)<d(X)$ then, as we have seen in proposition 2, $X$ has a non-constant meromorphic first integral, say $f\in\mathcal{M}(X)$. If $n=2$ and $f$ is purely meromorphic then $X$ is dicritical. ∎In the next section we will see examples of $X\in\mathcal{X}_{2}$ such that $r(X)=2<d(X)$ and $\mathcal{I}(X)=\mathbb{C}$. By corollary 2.1 these examples are dicritical. When $\mathcal{M}(X)\neq\mathbb{C}$ then $d(X)$, although finite, can be arbitrarily big, as shows the following example in any dimension $n\geq 2$:

Example 4**.**

Let $R=\sum_{j=1}^{n}z_{j}\frac{\partial}{\partial z_{j}}$ be the radial vector field and $X=(z_{1}...z_{n})^{k}.\,R$, where $k\in\mathbb{N}$. It can be verified that $C(X)=\mathcal{L}\,\oplus\,\mathcal{A}$, where $\mathcal{L}=$

[TABLE]

and

[TABLE]

Example 4 motivates the following question:

Problem 1**.**

Let $X\in\mathcal{X}_{n}$, where $X\not\equiv 0$. Does $d(X)=\infty$ implies that $\mathcal{I}(X)\neq\mathbb{C}$?

In dimension two problem 1 has a positive answer:

Theorem 1**.**

Let $X\in X_{2}$, $X\not\equiv 0$, be a vector field with $d(X)=\infty$. Then $\mathcal{I}(X)\neq\mathbb{C}$.

Proof. We will assume first that the vector space $D(X)=\{Y\in C(X)\,|\,Y\wedge X=0\}$ is infinite dimensional.

By corollary 2.1 we can assume that the origin is not an isolated singularity of $X$. In particular, we can write $X=g.\,T$, where $g\in\mathcal{O}_{2}$, $g(0)=0$ and $T\in\mathcal{X}_{2}$ has an isolated singularity at $0\in\mathbb{C}^{2}$.

If $Y\in D(X)\setminus\{0\}$ then $Y\wedge T=0$ and since [math] is an isolated singularity of $T$ we can write $Y=f_{Y}.\,T$, where $f_{Y}\in\mathcal{O}_{2}\setminus\{0\}$. Note that $f_{Y}/g\in\mathcal{M}(X)$ because $Y\in C(X)$ and $Y=\frac{f_{Y}}{g}\,X$. In particular, we can assume that $f_{Y}(0)=0$ for all $Y\in D(X)$, for otherwise $g/f_{Y}\in\mathcal{I}(X)\setminus\mathbb{C}$ and we are done. From now on, we will assume that $\mathcal{M}(X)\neq\mathbb{C}$ and, by contradiction, that $\mathcal{I}(X)=\mathbb{C}$.

Denote $\mathcal{V}(X):=\left<f_{Y}\,|\,Y\in D(X)\right>_{\mathbb{C}}$, the $\mathbb{C}$-vector subspace of $\mathcal{O}_{2}$ generated by the $f_{Y}$, $Y\in D(X)$. Since $dim(D(X))=\infty$ we must have $dim(\mathcal{V}(X))=\infty$. As a consequence, the set $\mu(X):=\{\mu(f_{Y},0)\,|\,Y\in D(X)\}\subset\mathbb{N}$ is unbounded.

In fact, let $j^{N}\colon\mathcal{O}_{2}\to\mathcal{O}_{2}/m_{2}^{N+1}$ be the $N^{th}$-jet map and $J^{N}=j^{N}|_{\mathcal{V}(X)}$ be its restriction to $\mathcal{V}(X)$. Note that $ker(J^{N})=\left<f_{Y}\in\mathcal{V}(X)\,|\,\mu(f_{Y},0)>N\right>_{\mathbb{C}}$. Since $dim(J^{N}(\mathcal{V}(X)))<\infty$ we get $dim(ker(J^{N}))=\infty$ for all $N\in\mathbb{N}$. Hence, $\mu(X)$ is unbounded.

Let $\{Y_{n}\}_{n}$ be a sequence of vector fields in $D(X)$ such that the sequence $\mu_{n}:=\mu(f_{Y_{n}},0)$ is increasing.

Recall that, by Seidenberg‘s resolution theorem [16], there exists a blowing-up process $\Pi\colon(M,E)\to(\mathbb{C}^{2},0)$ such that the strict transform $\Pi^{*}(\mathcal{F}_{T})$ of the foliation $\mathcal{F}_{T}$ is generically transverse to some irreducible component $D$ of the exceptional divisor $E$. In particular, at some generic point $p\in D$ there are local coordinates $(x,t)\colon(M,p)\to(\mathbb{C}^{2},0)$ such that:

• (i).

$x(p)=t(p)=0$ and $D_{p}=(x=0)$.

• (ii).

$\Pi^{*}(\mathcal{F}_{X})$ is defined at $p$ by $dt=0$.

• (iii).

$g\circ\Pi(x,t)=x^{k}.\,\widetilde{g}(x,t)$, where $x\not|\,\widetilde{g}$. We can choose $p$ in such a way that $\widetilde{g}$ is an unity.

• (iv).

$f_{Y_{n}}\circ\Pi(x,t)=x^{k_{n}}.\,\widetilde{f}_{n}(x,t)$, where $x\not|\,\widetilde{f}_{n}$.

Set $h_{n}:=\frac{f_{Y_{n}}}{g}\circ\Pi=x^{k_{n}-k}.\,\widetilde{h}_{n}$, where $\widetilde{h}_{n}=\widetilde{f}_{n}/\widetilde{g}$. Since $f_{Y_{n}}/g\in\mathcal{M}(T)$ we get $h_{n}\in\mathcal{M}(dt)$ and this implies

[TABLE]

However, it is known that if $f_{1},f_{2}\in\mathcal{O}_{2}$ and $\mu(f_{1},0)<\mu(f_{2},0)$ then $\mu(f_{1}\circ\Pi,p)<\mu(f_{2}\circ\Pi,p)$ and this implies that $\underset{n\to\infty}{\ell im}k_{n}=\infty$, a contradiction. Let us consider the general case: $d(X)=\infty$. We will assume also that $dim(D(X))<\infty$, so that $dim(C(X)/D(X)=\infty$. Fix a local coordinate system $(x,y)$ around $0\in\mathbb{C}^{2}$ and set $\mu=\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}$.

Given $Y\in C(X)\setminus D(X)$ we can write $Y\wedge X=f_{Y}.\,\mu$, where $f_{Y}\in\mathcal{O}_{2}$. Note that

[TABLE]

[TABLE]

[TABLE]

where $\nabla\,X=\frac{\partial X(x)}{\partial x}+\frac{\partial X(y)}{\partial y}$. Let $\{Y_{n}\}_{n}$ be a sequence in $C(X)\setminus D(X)$ such that the sequence $\mu_{n}:=\mu(f_{Y_{n}},0)$ is increasing and set $f_{n}:=f_{Y_{n}}$. From (2) we get

[TABLE]

Therefore, we can apply the same argument of the first case to the sequence of meromorphic first integrals $f_{n}/f_{1}$. ∎We finish this section with the following result:

Proposition 3**.**

Let $X\in\mathcal{X}_{n}$, $X\not\equiv 0$. If $r(X)=n$ then there exists a closed meromorphic $(n-1)$-form anihilating $X$.

Proof. Fix a local coordinate system $(z_{1},...,z_{n})$ around $0\in\mathbb{C}^{n}$ and let $\nu=dz_{1}\wedge...\wedge dz_{n}$ and $X=\sum_{j=1}^{n}X_{j}\frac{\partial}{\partial z_{j}}$. Set

[TABLE]

where $\widehat{dz_{j}}$ means the omission of $dz_{j}$ in the product. Since $r(X)=n$ there are vector fields $Y_{1},...,Y_{n}\in C(X)$ such that $Y_{1}\wedge...\wedge Y_{n}\not\equiv 0$. Set

[TABLE]

where $g\in\mathcal{O}_{n}$ and $\mu=\frac{\partial}{\partial z_{1}}\wedge...\wedge\frac{\partial}{\partial z_{n}}$. From (3) we get

[TABLE]

where $\nabla X=\sum_{j}\frac{\partial X_{j}}{\partial z_{j}}$. Since $Y_{j}\in C(X)$, $1\leq j\leq n$, we get from (4)

[TABLE]

$X(g)=\nabla X.\,g$. In particular

[TABLE]

Therefore, the meromorphic form $\frac{\omega}{g}$ is closed and satisfies $i_{X}\frac{\omega}{g}=0$. ∎

2.2. More general remarks.

We begin by the case in which $C(X)$ is a finite dimensional Lie algebra.

Proposition 4**.**

Let $X\in\mathcal{X}_{n}$ be a germ of holomorphic vector field, where $n\geq 2$. Suppose that $\mathcal{M}(X)=\mathbb{C}$, that is $X$ has no non constant meromorphic first integral. Then $C(X)$ is a finite dimensional Lie algebra of dimension $r(X)$. In particular, $d(X)=r(X)$.

Proof. Let $Y_{1},...,Y_{r}\,\in C(X)$, $r=r(X)$, be such that $Y_{1}\wedge...\wedge Y_{r}\not\equiv 0$. If $Y\in C(X)$ then there exist $f_{1},...,f_{r}\in\mathcal{M}_{n}$ such that

[TABLE]

In particular, we have:

[TABLE]

Since $\mathcal{M}(X)=\mathbb{C}$ we must have $f_{j}\in\mathbb{C}$, $1\leq j\leq r$. ∎

A particular case of proposition 4 is the following:

Corollary 2.2**.**

Let $X\in\mathcal{X}_{n}$, where $n\geq 2$. Suppose that $\mathcal{M}(X)=\mathbb{C}$ and that $C(X)\subset m_{n}^{2}.\,\mathcal{X}_{n}$. Then $C(X)$ is nilpotent.

Corollary 2.2 is a direct consequence of the following:

Proposition 5**.**

Let $\mathcal{L}\subset\mathcal{X}_{n}$ be a finite dimensional Lie algebra of germs of vector fields. Suppose that $\mathcal{L}\subset m_{n}^{2}.\,\mathcal{X}_{n}$. Then $\mathcal{L}$ is nilpotent.

Proof. Proposition 5 is a direct consequence of Engel-Lie theorem [9]. In fact, given $Y\in\mathcal{L}$ consider the operator

[TABLE]

Then the eigenvalues of $ad_{Y}$ are all zero, because

[TABLE]

Therefore $\mathcal{L}$ is nilpotent by the Engel-Lie theorem [9]. ∎

Remark 2.1**.**

In fact, proposition 5 can be generalized to finite dimensional Lie algebras $\mathcal{L}\subset m_{n}.\,\mathcal{X}_{n}$ such that the linear part $j^{1}\,\mathcal{L}$ is nilpotent.**

As a consequence we have:

Corollary 2.3**.**

Let $\mathcal{L}\subset m_{n}.\,\mathcal{X}_{n}$ be a finite Lie algebra and $J^{1}\colon\mathcal{L}\to gl(n,\mathbb{C})$ be the morphism $J^{1}(X)=DX(0)$. Then $ker(J^{1})=\mathcal{L}\cap m_{n}^{2}.\,\mathcal{X}_{n}$ is nilpotent.

Remark 2.2**.**

In example 4 $C(X)=\mathcal{L}\,\oplus\,\mathcal{A}$ is solvable but not nilpotent: in fact, as the reader can check $[\mathcal{L}\,\oplus\,\mathcal{A},\mathcal{L}\,\oplus\,\mathcal{A}]=[\mathcal{L},A]\subset\mathcal{A}$, and $[\mathcal{A},\mathcal{A}]=0$, but $[\mathcal{L},[\mathcal{L},\mathcal{A}]]=[\mathcal{L},\mathcal{A}]$.**

Another important result in this direction is the following (cf. [8] and [7]):

Theorem 2**.**

(R. Hermann, V. Guillemin, S. Sternberg)* Let $\mathcal{L}\subset m_{n}.\,\mathcal{X}_{n}$ be a semi-simple Lie algebra. Then $\mathcal{L}$ is linearizable, that is there exists $\phi\in Diff(\mathbb{C}^{n},0)$ such that $\phi_{*}(\mathcal{L})$ is a Lie algebra of linear vector fields.*

As a consequence, we have the following:

Corollary 2.4**.**

Let $X\in m_{n}.\,\mathcal{X}_{n}$ and assume that $\mathcal{M}(X)=\mathbb{C}$. Then, up to conjugacy the Levi-Malcev decomposition (see [10]) is of the form $C(X)=Rad\,\oplus\,\mathcal{L}$, where $\mathcal{L}$ is a semi-simple Lie algebra of linear vector fields and $Rad$ is the solvable radical of $C(X)$.

Some easy consequences in small dimension when $\mathcal{M}(X)=\mathbb{C}$:

If $r(X)=1$ then $C(X)=\mathbb{C}.\,X$.

• 2.

If $r(X)=2$ then $C(X)$ is abelian.

• 3.

If $r(X)=3$ then, either $C(X)$ is abelian, or $C(X)\simeq\mathbb{C}\,\bigoplus\,A$, where $A$ is the affine Lie algebra, or $C(X)$ is isomorphic to the Heisenberg algebra:

[TABLE]

Example 5**.**

Let $X$ be the "Poincaré-Dulac" vector field defined as

[TABLE]

Then $\mathcal{M}(X)=\mathbb{C}$ and $C(X)=\left<y^{m}\frac{\partial}{\partial x},y^{n}\frac{\partial}{\partial z},X\right>_{\mathbb{C}}$ is abelian of dimension three.**

Problem 2**.**

Is there $X\in\mathcal{X}_{3}$ such that $\mathcal{M}(X)=\mathbb{C}$, $r(X)=3$ the origin is an isolated singularity of $X$ and $C(X)$ is isomorphic to, either $\mathbb{C}\oplus\,A$ or to the Heisenberg algebra?

In his work on commuting vector fields [11] the second author shows the following result: Theorem [11]. Let $X\in\widehat{\mathcal{X}}_{2}$ with $X=\sum_{j\geq k}X_{j}$ its Taylor series, where $X_{j}$ is homogeneous of degree $j$ and $X_{k}\neq 0$. Assume that

• (a).

$k\geq 2$* and $X_{k}$ has an isolated singularity at $0\in\mathbb{C}^{2}$.*

• (b).

$X_{k}$* has no meromorphic first integral.*

Then $C(X)=\mathbb{C}.\,X$. A generalization of this result is the following:

Proposition 6**.**

Let $X\in\mathcal{X}_{n}$ with $X=\sum_{j\geq k}X_{j}$ its Taylor series, where $X_{j}$ is homogeneous of degree $j$ and $X_{k}\neq 0$. Assume that

• (a).

$k\geq 2$* and $X_{k}$ has an isolated singularity at $0\in\mathbb{C}^{2}$.*

• (b).

The foliation $\mathcal{F}_{X_{k}}$, induced by $X_{k}$ on $\mathbb{P}^{n-1}$, has no algebraic invariant set of dimension $\ell$, where $0<\ell<n-1$.

Then $C(X)=\mathbb{C}.\,X$.

Proof. The proof is based in the following:

Lemma 2.1**.**

Let $X_{k}$ be as in (a) and (b) of proposition 6. Let $Y_{\ell}\in\mathcal{X}_{n}$ be homogeneous of degree $\ell$ be such that $[X_{k},Y_{\ell}]=0$. Then:

• (i).

If $\ell\neq k$ then $Y_{\ell}=0$.

• (ii).

If $\ell=k$ then $Y_{k}=\lambda.\,X_{k}$, where $\lambda\in\mathbb{C}$.

Let us prove the proposition using lemma 2.1. Let $Y\in\mathcal{X}_{n}$ be such that $[X,Y]=0$ with Taylor series $Y=\sum_{j\geq\ell}Y_{j}$, where $Y_{j}$ is homogeneous of degree $j$ and $Y_{\ell}\neq 0$.

Since $[X_{r},Y_{s}]$ is homogeneous of degree $r+s-1$, we can write the Taylor series of $[X,Y]$ as $[X,Y]=\sum_{j\geq k+\ell-1}Z_{j}$, where $Z_{k+\ell-1}=[X_{k},Y_{\ell}]$. Therefore $[X_{k},Y_{\ell}]=0$, and by lemma 2.1 we get $\ell=k$ and $Y_{k}=\lambda.\,X_{k}$, where $\lambda\in\mathbb{C}^{*}$. Now, if we set $W=Y-\lambda.\,X$ then $[X,W]=0$ and

[TABLE]

by lemma 2.1. Hence, $Y=\lambda.\,X$ and $Y\in\mathbb{C}.\,X$. ∎Proof of lemma 2.1. First of all, hypothesis (b) of the proposition implies that if $A$ is an irreducible analytic subset of $(\mathbb{C}^{n},0)$ of dimension $dim(A)\geq 1$ and $X_{k}$-invariant then $A$ is a straigth line through the origin, corresponding to a singularity in $\mathbb{P}^{n-1}$ of $\mathcal{F}_{X_{k}}$. In particular, we have $\mathcal{M}(X_{k})=\mathbb{C}$.

Let $Y_{\ell}\neq 0$ be homogeneous of degree $\ell$ such that $[X_{k},Y_{\ell}]=0$. It is sufficient to prove that $X_{k}\wedge Y_{\ell}\equiv 0$. In fact, since $X_{k}$ has an isolated singularity at the origin this implies that $Y_{\ell}=f.\,X_{k}$, where $X_{k}(f)=0$. Since $\mathcal{M}(X_{k})=\mathbb{C}$ we get $\ell=k$ and $Y_{\ell}\in\mathbb{C}.\,X_{k}$.

Suppose by contradiction that $X_{k}\wedge Y_{\ell}\neq 0$. Let $R=\sum_{j=1}^{n}z_{j}\frac{\partial}{\partial z_{j}}$ be the radial vector field. Note that $X_{k}\wedge R\neq 0$, because otherwise [math] would not be an isolated singularity of $X_{k}$. We have two possibilities: $1^{st}$. $X_{k}\wedge Y_{\ell}\wedge R\equiv 0$. In this case, we can write

[TABLE]

where $f$ and $g$ are meromorphic. Note that $g\neq 0$. In fact, from (5) we get

[TABLE]

Again by (5) we have

[TABLE]

[TABLE]

where in the above relation we have used that $[R,X_{k}]=(k-1).\,X_{k}$.

Since $\mathcal{M}(X_{k})=\mathbb{C}$ we get $g\in\mathbb{C}^{*}$ and so $X_{k}(f)=(k-1)\,g\in\mathbb{C}^{*}$. Finally $X_{k}(f)=0$, because [math] is a singularity of $X_{k}$, a contradiction with $g\neq 0$. $2^{nd}$. $X_{k}\wedge Y_{\ell}\wedge R\not\equiv 0$. Let $A=\{z\in\mathbb{C}^{n}\,|\,X_{k}(z)\wedge Y_{\ell}(z)\wedge R(z)=0\}$. Observe first that $dim(A)\geq 2$: it is well known that the set of zeroes of a totally decomposable $r$-vector has dimension $\geq r-1$, unless it is empty. We assert that $A$ is $X_{k}$-invariant.

In fact, first of all we have

[TABLE]

If we write

[TABLE]

then the ideal defining $A$ is $\mathcal{I}(A)=\left<A_{ijk}\,|\,1\leq i<j<k\leq n\right>$. Finally, relation (6) implies that $X_{k}(\mathcal{I}(A))\subset\mathcal{I}(A)$, as the reader can check. Hence, $A$ is $X_{k}$-invariant and $dim(A)\geq 2$, a contradiction with hypothesis (b).

Therefore, $X_{k}\wedge Y_{\ell}=0$ which proves the lemma. ∎

2.3. Examples.

The aim of this section is to introduce some simple examples.

2.3.1. Linear vector fields on $\mathbb{C}^{2}$ and the saddle-node.

A non-zero linear vector field $X$ on $\mathbb{C}^{2}$ is one that can be written as $X=A(x,y)\frac{\partial}{\partial x}+B(x,y)\frac{\partial}{\partial y}$, where $A$ and $B$ are linear. According to Jordan’s normal form, after a linear change of variables and multiplication by a constant, it can be written in one of the following forms:

a. $X=X_{\lambda}=x\frac{\partial}{\partial x}+\lambda.\,y\frac{\partial}{\partial y}$, where $\lambda\in\mathbb{C}$ (semi-simple case).

b. $X=x\frac{\partial}{\partial y}$ (nilpotent case).

c. $X=x\frac{\partial}{\partial x}+(x+y)\frac{\partial}{\partial y}$. The saddle-node is more complicated. The germ $X\in\mathcal{X}_{2}$ has a saddle-node at $0\in\mathbb{C}^{2}$ when it has an isolated singularity at [math] and the linear part $DX(0)$ has one eigenvalue zero and the other non-zero. In this case, it is known that, after a formal change of variables and a multiplication by a constant, then the saddle-node can written as:

d. $X=x^{p+1}\,\frac{\partial}{\partial x}+y\,(1+\lambda\,x^{p})\,\frac{\partial}{\partial y}$ ,

where $p\in\mathbb{N}$ and $\lambda\in\mathbb{C}$. The formal invariants are the multiplicity $\mu=p+1$ and $\lambda\in\mathbb{C}$. The saddle-node in the formal normal form (d) is Liouvillian integrable, in the sense that the associated foliation can be defined also by a meromorphic closed form. The dual form, $\omega:=i_{X}dx\wedge dy$, has an integrating factor $d\left(\frac{1}{y.\,x^{p+1}}.\,\omega\right)=0$ :

[TABLE]

For this reason, $\lambda$ is called the residue. In the next table we specify $C(X)$, $r(X)$ and $d(X)$ in the above cases.

$\,\,\,\,X$ $C(X)$ $r(X)$ $d(X)$

$X_{1}=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ all linear vector fields 2 4

$X_{\lambda}\,,\,\lambda\notin\mathbb{Q}_{\leq 0}\cup\mathbb{N}\cup 1/\mathbb{N}$ $\mathbb{C}.\,\left<x\,\frac{\partial}{\partial x},y\,\frac{\partial}{\partial y}\right>$ 2 2

$X_{-p/q}\,,\,p,q\in\mathbb{N}$ $\mathbb{C}\{x^{p}.\,y^{q}\}.\,\left<x\,\frac{\partial}{\partial x},y\,\frac{\partial}{\partial y}\right>$ 2 $\infty$

$X_{0}=x\,\frac{\partial}{\partial x}$ $\mathbb{C}\{y\}.\,\left<x\,\frac{\partial}{\partial x},y\,\frac{\partial}{\partial y}\right>$ 2 $\infty$

$X_{n}\,,\,n\geq 2$ $\mathbb{C}.\left<x\frac{\partial}{\partial x},y\frac{\partial}{\partial y},x^{n}\frac{\partial}{\partial y}\right>$ 2 3

$x\,\frac{\partial}{\partial y}$ $\mathbb{C}\{x\}.\left<x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y},\frac{\partial}{\partial y}\right>$ 2 $\infty$

$x\,\frac{\partial}{\partial x}+(y+x)\frac{\partial}{\partial y}$ $\mathbb{C}.\left<x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y},x\frac{\partial}{\partial y}\right>$ 2 2

$x^{p+1}\frac{\partial}{\partial x}+y(1+\lambda\,x^{p})\frac{\partial}{\partial y}$ $\mathbb{C}.\left<x^{p}\left(x\frac{\partial}{\partial x}+\lambda\,y\frac{\partial}{\partial y}\right),y\,\frac{\partial}{\partial y}\right>$ 2 2

In the above table, examples 1, 3, 4, 5 and 6 have $d(X)>r(X)$. In examples 3, 4 and 6 we have $\mathcal{I}(X)\neq\mathbb{C}$. Examples 1 and 5 are dicritical and have purely meromorphic first integrals (see corollary 2.1 in § 2.1).

2.3.2. Commuting vector fields and the Cauchy-Riemann equations.

Let $Z=f(z)\frac{\partial}{\partial z}$ be a germ at $0\in\mathbb{C}$ of holomorphic vector field. We can split $Z$ into real and imaginary parts, $Z=X+i\,Y$, where $X$ and $Y$ are germs at $0\in\mathbb{R}^{2}$ of real analytic vector fields. Explicitly, if we write $z=x+i\,y$, $f(z)=u(x,y)+i\,v(x,y)$ and $\frac{\partial}{\partial z}=\frac{\partial}{\partial x}-i\,\frac{\partial}{\partial y}$, then $Z=X+i\,Y$, where $X=u(x,y)\frac{\partial}{\partial x}+v(x,y)\frac{\partial}{\partial y}$ and $Y=v(x,y)\frac{\partial}{\partial x}-u(x,y)\frac{\partial}{\partial y}$.

The Cauchy-Riemann equations imply that $X$ and $Y$ commute: $[X,Y]=0$. Since $X$ and $Y$ are real analytic we can consider their complexification, that we will denote again by $X$ and $Y$. This classical construction gives examples of non-homogeneous pairs of commuting vector fields on $(\mathbb{C}^{2},0)$ with $r(X)=2$. Up to conjugacy, the vector field $f(z)\frac{\partial}{\partial z}$ can be written as

a. $\frac{\partial}{\partial z}$, if $f(0)\neq 0$. In this case, $X$ and $Y$ are two commuting constant vector fields.

b. $\mu\,z\frac{\partial}{\partial z}$, $f(0)=0$ and $f^{\,\prime}(0)=\mu\neq 0$. In this case, $X$ and $Y$ are two linear vector fields with eigenvalues $\mu=a+i\,b$, $\overline{\mu}=a-i\,b$ and $\lambda=b+i\,a$, $\overline{\lambda}=b-i\,a$, respectively.

c. $X_{p,\,\lambda}=\frac{z^{p+1}}{1+\lambda\,z^{p}}\frac{\partial}{\partial z}$, if $f$ has multiplicity $p+1\geq 2$ at the origin. In this case, if $\lambda=0$ then $X$ and $Y$ are homogeneous and have a non-constant meromorphic first integral. If $\lambda\neq 0$ then $X$ and $Y$ have no meromorphic first integral.

In fact, the above construction can be generalized in any dimension. Given $Z\in\mathcal{X}_{n}$ we can write $Z=X+i\,Y$, where $X$ and $Y$ are germs of real analytic vector fields on $(\mathbb{R}^{2n},0)$. Their complexifications, called still $X$ and $Y$, are two commuting vector fields on $(\mathbb{C}^{2n},0)$. If $Z(0)=0$ then the distribution generated by $X$ and $Y$ has a singular locus of dimension $\geq 1$: the set $\{z\in(\mathbb{C}^{2n},0)\,|\,X(z)\wedge Y(z)=0\}$.

For instance, if $n=2$ then the Camacho-Sad theorem on the existence of an analytic separatrix of $Z$ (cf. [2]), gives a holomorphic separatrix $\gamma$ for $Z$ through $0\in\mathbb{C}^{2}$. The germ of surface obtained by complexification of $\gamma$, considered as real surface on $\mathbb{R}^{4}\simeq\mathbb{C}^{2}$, is invariant by both vector fields $X$ and $Y$. This motivates the following problem:

Problem 3**.**

Let $X$ and $Y$ be two germs at $0\in\mathbb{C}^{n}$, $n\geq 3$, of commuting vector fields such that $X\wedge Y\not\equiv 0$. Does there exists a germ a complex surface through $0\in\mathbb{C}^{n}$ which is simultaneously $X$ and $Y$ invariant.**

In the case $n=3$ problem 3 has positive answer [15]. However, we would like to note that in general, if $X\wedge Y\not\equiv 0$ then there is an analytic set of dimension $\geq 1$ invariant by both vector fields, $X$ and $Y$. This is a consequence of the following result:

Proposition 7**.**

Let $X\in\mathcal{X}_{n}$ and assume that the unique proper analytic subset of $(\mathbb{C}^{n},0)$ which is $X$-invariant is the origin $\{0\}$. Then $C(X)=\mathbb{C}.\,X$.

Proof. The hypothesis implies that $0\in Sing(X)$: if not, then the orbit of [math] by the local flow of $X$ has dimension one and is $X$-invariant. Note also that [math] is an isolated singularity of $X$.

Assume by contradiction that $C(X)\neq\mathbb{C}.\,X$. In this case, we have two possibilities:

• $1^{st}$.

$r(X)=1$ and $d(X)>1$.

• $2^{nd}$.

$r(X)>1$.

In the first case we have $\mathcal{I}(X)\neq\mathbb{C}$, which is not possible with the hypothesis. In the second case, there exists $Y\in C(X)$ such that $X\wedge Y\not\equiv 0$. Consider the analytic set

[TABLE]

Note that $dim(\Sigma)\geq 1$, because $0\in\Sigma$. On the other hand, if $z_{o}\neq 0$ and $z_{o}\in\Sigma$ then there are local coordinates $(U,x=(x_{1},...,x_{n}))$ at $z_{o}$ such that $x(z_{o})=0$ and:

• (a).

$X|_{U}=\frac{\partial}{\partial x_{1}}$, because $X(z_{o})\neq 0$.

• (b).

$Y(x)=\sum_{j=1}^{n}a_{j}(x_{2},...,x_{n})\frac{\partial}{\partial x_{j}}$, because $[X,Y]=0$.

Since $z_{o}\in\Sigma$ we get

[TABLE]

the analytic subset $A=(x_{2}=...=x_{n}=0)\subset U$ is contained in $\Sigma$ and is $X$-invariant, contradicting the hypothesis. ∎As an imediate consequence we have:

Corollary 2.5**.**

Let $X\in\mathcal{X}_{n}$ and assume that $d(X)\geq 2$. Then there exists a proper $X$-invariant analytic subset $\Sigma\subset(\mathbb{C}^{n},0)$ with $dim(\Sigma)\geq 1$.

3. The case of dimension two

In this section we study the case of dimension two. A crucial fact that will be used is that if $X\in\mathcal{X}_{2}$ has a non-constant holomorphic first integral, then there exists $f\in\mathcal{I}(X)$ such that $\mathcal{I}(X)=\mathbb{C}\{f\}$ (see [14]). The first integral $f$ is called a minimal first integral of $X$.

Another remark is that when $X\in\mathcal{X}_{2}$ has an isolated singularity at $0\in\mathbb{C}^{2}$, $r(X)=1$ and $d(X)\geq 2$, then $X$ has a non-constant holomorphic minimal first integral $f$, so that $C(X)=\mathbb{C}\{f\}.\,X$ and $d(X)=\infty$.

In fact, since $r(X)=1$ and $d(X)\geq 2$ there exists $Y\in C(X)$ such that $Y\notin\mathbb{C}.\,X$. But, $r(X)=1$ implies that $Y\wedge X=0$ and so $Y=g.\,X$ where $g\in\mathcal{O}_{2}\setminus\mathbb{C}$ by Hartogs extension theorem (here we use that $X$ has an isolated singularity at $0\in\mathbb{C}^{2}$). Note that $X(g)=0$ because

[TABLE]

3.1. The case $r(X)=2$ in dimension two.

We have proved in proposition 3 that if $X\in\mathcal{X}_{2}$ and $r(X)=2$ then $X$ is Liouvillian integrable: the foliation $\mathcal{F}_{X}$ is also defined by a germ of closed meromorphic 1-form. In other words, the dual form of $X$, $\omega:=i_{X}dx\wedge dy$, has an integrating factor $g$: $d\left(\frac{\omega}{g}\right)=0$. For instance, the saddle-node in the formal normal form is Liouvillian integrable.

In the general case, if the decomposition of the integrating factor $g$ into irreducible factors is $\Pi_{j=1}^{r}f_{j}^{k_{j}}$ then it is proved in [4] that:

[TABLE]

where

$f_{1},...,f_{r}\in\mathcal{O}_{2}$ are the poles of $\omega$, $\varphi\in\mathcal{O}_{2}$ and $f_{j}$ does not divides $\varphi$, $1\leq j\leq r$.

• 2.

$\lambda_{j}\in\mathbb{C}$ is the residue of $\frac{\omega}{f}$ along $f_{j}$, $1\leq j\leq r$.

• 3.

$k_{j}\in\mathbb{N}$ is the multiplicity of $\frac{\omega}{f}$ at the pole $f_{j}$, $1\leq j\leq r$.

The foliation $\mathcal{F}_{X}$ is said to be logarithmic if $\varphi/f_{1}^{k_{1}-1}...f_{r}^{k_{r}-1}$ is holomorphic, or equivalently $k_{1}=...=k_{r}=1$.

Remark 3.1**.**

Each irreducible component of the pole $f$ of $\frac{\omega}{f}$ defines a separatrix $\Gamma_{j}:=(f_{j}=0)$ of $\mathcal{F}_{X}$.

Another observation is that the "multivalued" function $\sum_{j}\lambda_{j}\,log(f_{j})+\varphi/f^{k_{1}-1}...f_{r}^{k_{r}-1}$ is a first integral of $X$.**

Suppose now that $X_{1}$ and $X_{2}$ are two holomorphic germs on $(\mathbb{C}^{2},0)$ of commuting vector fields such that $X_{1}\wedge X_{2}\not\equiv 0$.

Claim 3.1**.**

There are unique closed meromorphic 1-forms $\alpha_{1}$ and $\alpha_{2}$ such that $\alpha_{i}(X_{j})=\delta_{ij}$, $i,j=1,2$, where $\delta_{ij}=1$ if $i=j$ and [math] otherwise.

Proof. As we have seen in the proof of proposition 3, if we set $X_{1}\wedge X_{2}=g\frac{\partial}{\partial z_{1}}\wedge\frac{\partial}{\partial z_{2}}$, $\omega_{1}=i_{X_{2}}dz_{1}\wedge dz_{2}$ and $\omega_{2}=-i_{X_{1}}dz_{1}\wedge dz_{2}$, then the forms $\alpha_{1}:=\frac{\omega_{1}}{g}$ and $\alpha_{2}:=\frac{\omega_{2}}{g}$ are closed. As the reader can check, we have also $\alpha_{i}(X_{j})=\delta_{ij}$, $1\leq i,j\leq 2$. ∎As a consequence, we have the following:

Proposition 8**.**

Given $X\in\mathcal{X}_{2}\setminus\{0\}$ we have three possibilities:

• (a).

$r(X)=d(X)=1$. In this case, $C(X)=\mathbb{C}.\,X$.

• (b).

$d(X)\geq 2$* and $r(X)=1$. In this case, $X$ has a non-constant meromorphic first integral.*

• (c).

$r(X)=2$. In this case $X$ is Liouvillian integrable.

Corollary 3.1**.**

Let $X\in\mathcal{X}_{2}$ be a saddle-node. Then $X$ is holomorphically normalisable if, and only if, $r(X)=2$.

Corollary 3.2**.**

Let $X\in\mathcal{X}_{2}$ with $DX(0)=x\frac{\partial}{\partial x}+\lambda\,y\frac{\partial}{\partial y}$, with $\lambda\in\mathbb{C}^{*}\setminus\mathbb{Q}$. Then $X$ is linearizable if, and only if, $r(X)=2$.

Another interesting consequence is the following:

Corollary 3.3**.**

Let $X\in\mathcal{X}_{2}$ with $r(X)=2$. Let $\Pi\colon(M,E)\to(\mathbb{C}^{2},0)$ be the Seidenberg resolution of singularities of $X$ and $\widetilde{\mathcal{F}}_{X}$ be the strict transform of the foliation defined by $\Pi^{*}(X)$ (cf. [16] and § 3.2). Then the holonomy of $\widetilde{\mathcal{F}}_{X}$ in any non-dicritical irreducible component of $E$ is abelian.

3.2. Germs of vector fields with just one irreducible separatrix.

In the case of dimension two, an important class of foliations on $(\mathbb{C}^{2},0)$ are the so-called "generalized curves" defined in [3]. These foliations have no saddle nodes in their reduction of singularities. According to Seidenberg‘s resolution theorem [16], given a germ at $0\in\mathbb{C}^{2}$ of foliation $\mathcal{F}$ there exists a blowing-up procedure $\Pi\colon(M,D)\to(\mathbb{C}^{2},0)$ such that the strict transform $\Pi^{*}(\mathcal{F})$ of $\mathcal{F}$ has only reduced singularities on the exceptional divisor $D$. We say that a singularity $p\in D$ of $\Pi^{*}(\mathcal{F})$ is reduced if the germ at $p$ of the foliation is represented by a vector field $Y$ such that its linear part $DY(p)$ at $p$ has eigenvalues $\lambda_{1},\lambda_{2}$, where:

• •

either $\lambda_{1}=0$ and $\lambda_{2}\neq 0$ (or vice-versa). In this case, the singularity is a saddle-node,

• •

or $\lambda_{1}.\lambda_{2}\neq 0$ and $\lambda_{2}/\lambda_{1}\notin\mathbb{Q}_{+}$.

We will say that $X\in\mathcal{X}_{2}$ is a generalized curve (briefly G.C) if the associated foliation has no saddle-node in its resolution of singularities. In this case, we have the following result:

Theorem 3**.**

Let $X\in\mathcal{X}_{2}$ be a G.C vector field with an isolated singularity at $0\in\mathbb{C}^{2}$ and with just one separatrix. Then $r(X)=1$. In particular, $C(X)=\mathcal{I}(X).\,X$.

When a germ $X\in\mathcal{X}_{2}$ has a holomorphic first integral $f$, whoose decomposition into irreducible factors is $f=\Pi_{j=1}^{r}f_{j}^{k_{j}}$, then the separatrices of $X$ through $0\in\mathbb{C}^{2}$ are the curves $(f_{j}=0)$, $1\leq j\leq r$. In this case, $X$ is non-dicritical. In particular, if $X$ has an irreducible holomorphic first integral, say $f$ , then $(f=0)$ is the unique irreducible separatrix of $X$. In this case, we have the following:

Corollary 3.4**.**

Let $X\in\mathcal{X}_{2}$ and assume that [math] is an isolated singularity of $X$ and that $X$ has an irreducible first integral $f\in\mathcal{O}_{2}$. Then $r(X)=1$ and $C(X)=\mathbb{C}\{f\}.\,X$. Moreover, there exists a local coordinate system $(x,y)$ around $0\in\mathbb{C}^{2}$ such that $X=H_{f}$, where $H_{f}=f_{y}\frac{\partial}{\partial x}-f_{x}\frac{\partial}{\partial y}$ is the Hamiltonian of $f$.

Before proving theorem 3 we will see two examples in which the conclusion of theorem 3 is not true.

Example 6**.**

If $X$ is non singular, $X(0)\neq 0$, then in some coordinate system $(x,y)$ around [math] we have $X=\frac{\partial}{\partial x}$. On the other hand, it is not difficult to see that

[TABLE]

Example 7**.**

If $f$ is not irreducible then the theorem is not true in general. For instance, if $X=x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}=H_{x.y}$ then $x.y$ is a minimal first integral of $X$: $\mathcal{I}(X)=\mathbb{C}\{x.y\}$, but

[TABLE]

which is not of the form $\mathbb{C}\{f\}.\,X$.**

Proof of theorem 3. Observe first that the irreducibility of $f$ implies that if $\mathcal{I}(X)\neq\mathbb{C}$ then, modulo an unit, we can assume that $f$ is a minimal first integral of $X$ (see [14]). Since $f\in\mathcal{I}(X)$ the curve $\Gamma:=(f=0)$ is $X$-invariant (it is a separatrix of $X$). In fact, we have the following:

Lemma 3.1**.**

If $Y\in C(X)$ then $\Gamma$ is $Y$-invariant. Moreover, there exists $\lambda\in\mathbb{C}$ such that $Y|_{\Gamma}=\lambda.\,X|_{\Gamma}$.

Proof. We will assume $Y\not\equiv 0$. In some coordinate system $(x,y)$ around $0\in\mathbb{C}^{2}$ we can write $X=X_{1}\frac{\partial}{\partial x}+X_{2}\frac{\partial}{\partial y}$ and

[TABLE]

and $g(0)=0$. $1^{st}$ case. $g\equiv 0$. In this case, we have $Y=\phi.\,X$, where $\phi\in\mathcal{O}_{2}$, because $X$ has an isolated singularity at [math]. Since $Y\in C(X)$ we get $Y(\phi)=0$ and $\phi\in\mathcal{I}(X)$. If $\phi\in\mathbb{C}^{*}$ we are done. If $\phi$ is non-constant then $\mathcal{I}(X)\neq\mathbb{C}$ and we can assume that $f\in\mathcal{I}(X)$.

From $[X,Y]=0$ we get

[TABLE]

$\Gamma$ is $Y$-invariant. $2^{nd}$ case. $g\not\equiv 0$. If $L$ denotes the Lie derivative then

[TABLE]

which implies $g\,|\,X(g)$ and so $g$ is $X$-invariant and each irreducible component of $g$ is a separatrix of $X$. Since $\Gamma$ is the unique separatrix of $X$ we must have $g=u.\,f^{k}$, where $u\in\mathcal{O}_{2}^{*}$ and $k\geq 1$. Analogously,

[TABLE]

$\Gamma$ is $Y$-invariant. Let $\gamma(t)=(x(t),y(t))$ be a Puiseux‘s parametrization of $\Gamma$. Since $\Gamma$ is $X$ and $Y$-invariant we can write $\gamma^{*}(X)=\phi(t)\frac{\partial}{\partial t}$ and $\gamma^{*}(Y)=\varphi(t)\frac{\partial}{\partial t}$. Note that $\phi(0)=0$, but $\phi\not\equiv 0$, for otherwise [math] would not be an isolated singularity of $X$. On the other hand, $[X,Y]=0$ implies

[TABLE]

[TABLE]

which proves lemma 3.1. ∎End of the proof of theorem 3. Let $Z_{1}=Y-\lambda.\,X$. From $Z_{1}|_{\Gamma}=0$ we get $Z_{1}=f^{k_{1}}.\,Y_{1}$, where $Y_{1}\in\mathcal{X}_{2}$ and $k_{1}\geq 1$. We have two possibilities: $1^{st}$* case:* $\mathcal{I}(X)\neq\mathbb{C}$. In this case, we can assume that $\mathcal{I}(X)=\mathbb{C}\{f\}$. Since $Z_{1}\in C(X)$ and $f\in\mathcal{I}(X)$ we get $Y_{1}\in C(X)$. From lemma 3.1 we get again $Y_{1}=\lambda_{1}.\,X+f^{k_{2}}.\,Y_{2}$, where $\lambda_{1}\in\mathbb{C}$, $Y_{2}\in\mathcal{X}_{2}$ and $k_{2}\geq 1$, so that

[TABLE]

where $[X,Y_{2}]=0$. Using the above argument inductively we get a formal series $g(z)=\sum_{j=0}^{\infty}\lambda_{j}\,z^{j}$ such that $Y=g(f)\,.X$. Since $X$ and $Y$ are holomorphic, it is clear that the series $g$ is convergent and so $Y\in\mathbb{C}\{f\}.\,X$ and $r(X)=1$.

$2^{nd}$* case:* $\mathcal{I}(X)=\mathbb{C}$. We have two sub-cases:

2.a. $X\wedge Y\equiv 0$. In this case, we have $Y=h.\,X$ where $h\in\mathcal{O}_{2}$, because $X$ has an isolated singularity at $0\in\mathbb{C}^{2}$. Here $h\in\mathcal{I}(X)$ and since $\mathcal{I}(X)=\mathbb{C}$ we get $h\in\mathbb{C}$ and so $h=\lambda$ and $Y=\lambda.\,X\in\mathbb{C}.\,X$.

2.b. $X\wedge Y\not\equiv 0$. We will see that $\mathcal{I}(X)\neq\mathbb{C}$, a contradiction.

In fact, let $Y\in C(X)$ be such that $X\wedge Y\neq 0$. We assert that the set $\Gamma:=\{z\in(\mathbb{C}^{2},0)\,|\,X(z)\wedge Y(z)=0\}$ is $X$-invariant:

The set $\Gamma$ is an analytic curve through $0\in\mathbb{C}^{2}$: if we fix local coordinates $(x,y)$ around [math], then $X\wedge Y=g.\,\mu$, where $\mu:=\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}$, and $\Gamma=(g=0)$. On the other hand,

[TABLE]

[TABLE]

where $\nabla X=\frac{\partial X(x)}{\partial x}+\frac{\partial X(y)}{\partial y}$. Therefore, $\Gamma$ is $X$-invariant and is the unique separatrix of $X$. In particular, $\Gamma=(f=0)$ and after multiplying $g$ by an unit we can assume $g=f^{k}$. Let $\alpha$ and $\beta$ be the meromorphic closed 1-forms with $\alpha(X)=\beta(Y)=0$ and $\alpha(Y)=\beta(X)=1$. As we have seen before $\alpha=\frac{1}{f^{k}}\,i_{X}\nu$ and $\beta=\frac{1}{f^{k}}\,i_{Y}\nu$, $\nu=dx\wedge dy$. In particular, we can write [4]

[TABLE]

where $\lambda\in\mathbb{C}$, $H\in\mathcal{O}_{2}$, and, either $k=1$ and $\lambda\neq 0$, or $k>1$. When $k>1$ we will assume also that $f$ does not divides $H$.

If $\lambda\neq 0$ and $k=1$ then $\widetilde{f}:=f.\,e^{\lambda^{-1}H}$ is an irreducible holomorphic first integral of $X$, because

[TABLE]

If $k>1$ then $H\not\equiv 0$ and we can write $H=h_{0}+h$, where $h_{0}=H(0)$. We have four possilities:

• $1^{st}$.

$h_{0}=0$ and $\lambda=0$. In this case, $X$ is dicritical, because $f$ does not divides $H=h$ and all curves of the form $(f^{k-1}-c\,h=0)$, $c\in\mathbb{C}$, are $X$-invariant, so that $X$ has more than one separatrix.

• $2^{nd}$.

$h_{0}\neq 0$ and $\lambda=0$. In this case, there exists $\phi\in\mathcal{O}_{2}^{*}$ such that $\phi^{k-1}=H$ and $\widetilde{f}:=f/\phi$ is an irreducible holomorphic first integral of $X$.

• $3^{rd}$.

$h_{0}\neq 0$ and $\lambda\neq 0$. In this case, we have

[TABLE]

We see that $f^{k}\,\alpha$ is the pull-back by the morphism $(x,y)\mapsto(f,h)=(u,v)$ of the 1-form

[TABLE]

Since $\widetilde{\alpha}(0)=-(k-1)\,h_{0}\,du\neq 0$, by Frobenius theorem $\widetilde{\alpha}$ has a holomorphic first integral $F(u,v)$ of the form $(u,v)\mapsto F(u,v)=u+h.o.t.$, so that $X$ has a first integral of the form $F(g,h)=f+...$. Since $(f=0)$ is a separatrix of $X$, we must have $F(f,h)=u.\,f:=\widetilde{f}$, $u\in\mathcal{O}_{2}^{*}$, so that $X$ has an irreducible holomorphic first integral.

$4^{th}$. $h_{0}=0$ and $\lambda\neq 0$. In this case (8) can be written as

[TABLE]

and $f^{k-1}.\,\alpha$ is the pull-back of the 1-form

[TABLE]

by the map $(x,y)\mapsto(f,h)=(u,v)$. The dual vector field of the above 1-form is

[TABLE]

which is in the Poincaré-Dulac normal form. If $\lambda\neq 0$ then this vector field has a saddle-node in its redution of singularities [14]. This implies that $X$, the original vector field, has the same property and so is not G.C. Hence, in all possible cases, $X$ has an irreducible holomorphic first integral. ∎Proof of corollary 3.4. Since $X$ has a holomorphic first integral $f$ it is G.C (cf. [3]). Since $f$ is irreducible $X$ has an unique separatrix. Therefore, by theorem 3 we have $C(X)=\mathbb{C}\{f\}.\,X$. It remains to prove that in some coordinate system aroud $0\in\mathbb{C}^{2}$ we have $X=H_{f}$.

Fix a local coordinate system $(u,v)$. Since $X(f)=0$ and $0\in\mathbb{C}^{2}$ is an isolated singularity of $X$ and of $df$, there exists an unity $\phi\in\mathcal{O}_{2}^{*}$ such that $\phi.\,i_{X}\,du\wedge dv=df$. If we consider the change of variables $x=u$ and $y=\varphi.\,v$, where $\varphi\in\mathcal{O}_{2}^{*}$, then we get

[TABLE]

The p.d.e. $v.\,\varphi_{v}+\varphi=\phi$ has a solution $\varphi\in\mathcal{O}_{2}^{*}$. Hence in the new coordinate system we have

[TABLE]

Remark 3.2**.**

Given $f,g\in\mathcal{O}_{2}\setminus\{0\}$, with $f(0)=g(0)=0$, it is easy to check that $[H_{f},H_{g}]=0$ if, and only if, the jacobian determinant of $det\,J(f,g)$ is a constant $\lambda$. When $\lambda\neq 0$ then the map $(f,g)\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{2},0)$ is a germ of biholomorphism. In particular, $f$ is a submersion and $H_{f}(0)\neq 0$.

If $\lambda=0$ then, after [14], we can write $f=\phi(h)$ and $g=\psi(h)$, where $\phi,\psi\in\mathcal{O}_{1}$ and $h$ is a primitive first integral of $H_{f}$. We are essentially in the situation of theorem 3: $H_{f},H_{g}\in C(H_{h})$ and $r(H_{f})=1$.

In contrast, in dimension $n\geq 3$ we have examples of $f\in\mathcal{O}_{n}$ irreducible, but with $r(H_{f})>1$. This type of example can be constructed as follows: let $\mathbb{C}^{n}=E_{1}\oplus...\oplus E_{s}$ be a linear decomposition of $\mathbb{C}^{n}$, $X_{i}\in\mathcal{X}(E_{i},0)$, $i=1,...,s$, and set $X=X_{1}+...+X_{s}\in\mathcal{X}_{n}$. Then it is clear that

[TABLE]

where $\mathcal{I}(X_{j})\subset\mathcal{O}(E_{j},0)$ is the subring of first integrals of $X_{j}$, $1\leq j\leq s$.

For instance, let $\mathbb{C}^{2n}=\mathbb{C}^{2}\oplus...\oplus\mathbb{C}^{2}$ with the sympletic form $\Omega=dx_{1}\wedge dy_{1}+...+dx_{n}\wedge dy_{n}$. For each $j=1,...,n$ let $f_{j}=f_{j}(x_{j},y_{j})$ and $f=f_{1}+...+f_{n}$, so that the associated hamiltonian is given by $df=i_{H_{f}}\Omega$,

[TABLE]

The ring $\mathcal{R}:=\mathbb{C}\{f_{1},...,f_{n}\}$ is a subring of $\mathcal{I}(H_{f})$ and $C(H_{f})$ contains

[TABLE]

In fact, given $h_{1},h_{2}\in\mathcal{R}$ then $[H_{h_{1}},H_{h_{2}}]=0$.**

3.3. Generalized curves.

In this section we will study commuting G.C vector fields of a particular type.

Definition 1**.**

We say that $X\in\mathcal{X}_{2}$ has an isolated non-rational singularity at $0\in\mathbb{C}^{2}$ if its linear part $DX(0)$ has non-vanishing eigenvalues, $\lambda_{1},\lambda_{2}$, with $\lambda_{2}/\lambda_{1}\notin\mathbb{Q}$. When the quotient $\lambda_{2}/\lambda_{1}\in\mathbb{Q}$ we will say that [math] is an isolated rational singularity.

We say that $Y\in\mathcal{X}_{2}$ has a non-rational singularity at $0\in\mathbb{C}^{2}$ if $Y=g.\,X$, where $g\in\mathcal{O}_{2}$ and $X$ has an isolated non-rational singularity at [math].

If $X\in\mathcal{X}_{2}$ has a non-reduced singularity at $0\in\mathbb{C}^{2}$ then we say that it is of non-rational type if it has a non-rational singularity in its reduction of singularities.**

Our main result in this section is the following:

Theorem 4**.**

Let $X\in\mathcal{X}_{2}$ be a G.C (non-dicritical) with an isolated and non-rational singularity at $0\in\mathbb{C}^{2}$. If $DX(0)$ is nilpotent then $C(X)=\mathbb{C}.\,X$.

Proof. We need a lemma.

Lemma 3.2**.**

Let $Z\in\mathcal{X}_{2}$ with a singularity at $0\in\mathbb{C}^{2}$ and $\Pi\colon(\widetilde{\mathbb{C}}^{2},D)\to(\mathbb{C}^{2},0)$ be a blowing-up with exceptional divisor $D\simeq\mathbb{P}^{1}$. If $DZ(0)=0$ then:

• (a).

$\Pi^{*}(Z)|_{D}\equiv 0$.

• (b).

For any singularity $p\in D$ of the strict transform of $\Pi^{*}(Z)$ then $D\Pi^{*}(Z)(p)=0$.

Proof. The proof relies in the Seidenberg’s reduction process of $Z$ [16]. Write the Taylor series of $Z$ as

[TABLE]

where $Z_{j}$ is homogeneous of degre $j$ and $Z_{\nu}$ is the first non-zero jet of $Z$. Since $DZ(0)=0$ we have $\mu(Z,0)=\nu\geq 2$. Locally, in suitable coordinates, $\Pi$ is a quadratic map $\Pi(x,t)=(x,t.\,x)$ and $\Pi^{-1}(0)=(x=0)$. If $Z=A(x,y)\frac{\partial}{\partial x}+B(x,y)\frac{\partial}{\partial y}$ then

[TABLE]

If $Z=\sum_{j\geq\nu}Z_{j}$, as before, where $Z_{j}=A_{j}(x,y)\frac{\partial}{\partial x}+B_{j}(x,y)\frac{\partial}{\partial y}$ then we get $\pi^{*}(Z)=\sum_{j\geq\nu}\Pi^{*}(Z_{j})$, where

[TABLE]

From the above formula, we have two possibilities:

• $1^{st}$.

$B_{\nu}(1,t)-t.\,A_{\nu}(1,t)\not\equiv 0$. We see that $\Pi^{*}(Z)=x^{\nu-1}.\,\widetilde{Z}$, where $\widetilde{Z}$ has isolated singularities on the exceptional divisor $D$, which is $\widetilde{Z}$-invariant. This is case of a non-dicritical blow-up. The foliation on $\widetilde{\mathbb{C}}^{2}$ induced by $\widetilde{Z}$ is called the strict transform of $Z$ and denoted as $\widetilde{\mathcal{F}}_{Z}$. Observe also that

• 1.1.

$Sing(\widetilde{\mathcal{F}}_{Z})\cap D\neq\emptyset$.

• 1.2.

If $p\in Sing(\widetilde{\mathcal{F}}_{Z})\cap D$ then the algebraic multiplicity of $\Pi^{*}(Z)$ at $p$, $\mu(\Pi^{*}(Z),p)\geq\nu$.

• $2^{nd}$.

$B_{\nu}(1,t)-t.\,A_{\nu}(1,t)\equiv 0$. We see that $\Pi^{*}(Z)=x^{\nu}.\,\widetilde{Z}$, where $\widetilde{Z}$ has isolated singularities on $D$, which is not $\widetilde{Z}$-invariant. This is the case of a dicritical blow-up. Note that this happens if, and only if, $X_{\nu}$ is colinear with the radial vector field $x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$.

This proves lemma 3.2. ∎

Let us continue the proof of theorem 4. In the non-dicritical hypothesis of theorem 4 we are excluding the $2^{nd}$ possibility in any step of the reduction process for $X$. Let $\Pi\colon(M,D)\to(\mathbb{C}^{2},0)$ be the reduction of the vector field $X$, where $D:=\Pi^{-1}(0)$ is the exceptional divisor. Let $D=\bigcup_{j=1}^{r}D_{j}$ be the decomposition of $D$ into irreducible components. Denote by $\widetilde{\mathcal{F}}_{X}$ the foliation induced by $\Pi^{*}(X)$. We will assume that $D_{1}$ is strict transform by $\Pi$ of the divisor obtained at the first step of the resolution.

Remark 3.3**.**

When $DX(0)\neq 0$ is nilpotent we can assume $DX(0)=y\frac{\partial}{\partial x}$. In this case, in the first blow-up $\Pi_{1}\colon(\widetilde{\mathbb{C}}^{2},D_{1})\to(\mathbb{C}^{2},0)$ we find just one singularity on the divisor $D_{1}$, the point $p\in D_{1}$ corresponding to the direction $y=0$. The vector field $\Pi_{1}^{*}(X)$ is not identically zero on $D_{1}$, but $\mu(\Pi_{1}^{*}(X),p)=2$ (see [14]). The point $p$ is not a simple singularity for the foliation defined by $\Pi_{1}^{*}(X)$ and so we have to blow-up more times in the resolution process. If $X$ is G.C and at each step of the resolution we blow-up only at singularities of the strict transform then lemma 3.2 implies that $\Pi^{*}(X)|_{D_{j}}\equiv 0$ for all $j\geq 2$. As a consequence, we have the following:

Corollary 3.5**.**

Let $D_{k}$ be an irreducible component of $D$ where there is a non-rational singularity of $\widetilde{\mathcal{F}}_{X}$. Then $\Pi^{*}(X)|_{D_{k}}\equiv 0$.

In the above resolution if $i\neq j$ then, either $D_{i}\cap D_{j}=\emptyset$, or $D_{i}$ cuts $D_{j}$ transversely in just one point. Denote as $\widetilde{\mathcal{F}}_{X}$ the strict transform of the foliation induced by the vector field $\Pi^{*}(X)$. As a consequence of the above computation, we can conclude that:

• (i).

The strict transform $\widetilde{\mathcal{F}}_{X}$ has singularities in all components $D_{j}$ of the exceptional divisor (see [1] and [14]). This follows from the hypothesis that $X$ is non-dicritical.

• (ii).

The vector field $\Pi^{*}(X)$, which is holomorphic on $M$, vanishes identically along all components $D_{j}$ of exceptional divisor, except perhaps at $D_{1}$.

Claim 3.2**.**

Let $p\in D$ be a non-rational singularity of $\widetilde{\mathcal{F}}_{X}$. Then:

• (a).

There are local coordinates around $p$, $(x,y)\colon(M,p)\to(\mathbb{C}^{2},0)$, such that the first non-zero jet of $\Pi^{*}(X)$ is of the form $x^{p}\,y^{q}\,\left(x\frac{\partial}{\partial x}+\lambda\,y\frac{\partial}{\partial y}\right)$, where $\lambda\notin\mathbb{Q}$ and $p+q\geq 1$.

• (b).

The germ of $\Pi^{*}(X)$ at $p$ is formally conjugated to $x^{p}\,y^{q}\,\left(x\frac{\partial}{\partial x}+\lambda\,y\frac{\partial}{\partial y}\right)$.

Proof. First of all, we have two possibilities for $p\in D$:

• (1).

$p\in D_{j}$ and $p\notin D_{i}$ if $i\neq j$.

• (2).

$p\in D_{i}\cap D_{j}$, with $i\neq j$, and $p\notin D_{s}$, $s\neq i,j$.

Note that there are local coordinates around $p$, $(x,y)\colon(U,p)\to(\mathbb{C}^{2},0)$, in which in the first case we have $D\cap U=D_{j}\cap U=(x=0)$ and in the second case we have $D\cap U=(D_{i}\cup D_{j})\cap U=(x.\,y=0)$.

It follows from (ii) above that $\Pi^{*}(X)$ vanishes in a certain order $\geq 1$ along each $D_{i}\subset D$, so that, in both cases we can write $\Pi^{*}(X)=x^{p}\,y^{q}\,Z$, where $p+q\geq 1$ and $Z$ has an isolated singularity at $0\in\mathbb{C}^{2}$. In case (1) we have $p\geq 1$ and $q=0$, whereas in case (2) we have $p,q\geq 1$. Since $p$ is a non-rational singularity of $\widetilde{\mathcal{F}}_{X}$, then $det(DZ(0))\neq 0$ and the eigenvalues $\lambda_{1},\lambda_{2}$ of $DZ(0)$ satisfy $\lambda_{2}/\lambda_{1}\notin\mathbb{Q}$. In particular, $Z$ has two smooth and transverse separatrices through $p$ (see [14]). In case (2) necessarily these separatrices are contained in $D_{i}\cup D_{j}=(x.\,y=0)$, whereas in case (1), one of the separatrices is contained in the exceptional divisor $(x=0)$ and the other, after a holomorphic change of variables, we can assume that is $(y=0)$. In both cases, after dividing $Z$ by $\lambda_{1}$, we can assume that the first jet of $Z$ at $0\in\mathbb{C}^{2}$ is of the form $x\frac{\partial}{\partial x}+\lambda\,y\frac{\partial}{\partial y}$, $\lambda=\lambda_{2}/\lambda_{1}$. In this case, by Poincaré’s linearization theorem [13], $Z$ is formally conjugated to its linear part $x\frac{\partial}{\partial x}+\lambda\,y\frac{\partial}{\partial y}$ by a formal $\Phi\in\widehat{Diff}(\mathbb{C}^{n},0)$ such that $\Phi(u,v)=(u\,\Phi_{1}(u,v),v\,\Phi_{2}(u,v))=(x,y)$: $\Phi^{*}(Z)=u\frac{\partial}{\partial u}+\lambda\,v\frac{\partial}{\partial v}:=\widehat{Z}$. In this case, we must have

[TABLE]

where $\phi\in\widehat{\mathcal{O}}_{2}^{*}$, and $\phi(0)=1$. Consider now a formal diffeomorphism of the form $\Psi(x,y)=(e^{\alpha}.\,x,e^{\lambda\,\alpha}.\,y)=(u,v)$, where $\alpha(0)=0$. With a straightforward computation we have

[TABLE]

so that

[TABLE]

This reduces the proof of the claim to find a solution of the differential equation

[TABLE]

If we set $w=e^{(p+\lambda\,q)\alpha}$ then the above differential equation becomes

[TABLE]

Since $\lambda\notin\mathbb{Q}$ the linear operator

[TABLE]

is surjective, as the reader can check. Hence, the differential equation (9) has a formal solution. This proves the claim. ∎Let us suppose by contradiction that $C(X)\neq\mathbb{C}.\,X$ and let $Y\in C(X)\setminus\mathbb{C}.\,X$. We have two possibilities: $1^{st}$. $Y\wedge X\equiv 0$. In this case, since $X$ has an isolated singularity at [math], we must have $Y=f.\,X$, where $f\in\mathcal{O}_{2}$ is a non-constant first integral of $X$. But, when $X$ has a non-constant first integral all singularities of the strict transform $\widetilde{\mathcal{F}}_{X}$ are rational (see [14]).

$2^{nd}$. $Y\wedge X\not\equiv 0$. Set $Y\wedge X=f.\,\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}$, $f\in\mathcal{O}_{2}$, $f(0)=0$. As we have seen in the proof of theorem 3 in § 3.2, the curve $(f=0)$ is $X$ and $Y$-invariant: if $f=\Pi_{j=1}^{r}f_{j}^{k_{j}}$ is the decomposition of $f$ into irreducible factors, then the curves $\Gamma_{j}=(f_{j}=0)$, $1\leq j\leq r$, are all simultaneously $X$ and $Y$-invariant. Moreover, if $\alpha=\frac{1}{f}\,i_{Y}dx\wedge dy$ and $\beta=-\frac{1}{f}\,i_{X}dx\wedge dy$ then $\alpha(X)=\beta(Y)=1$, $\alpha(Y)=\beta(X)=0$ and $d\alpha=d\beta=0$. The curve $\Gamma:=(f_{1}...f_{r}=0)$ is called the reduced separatrix of $X$. We denote $\Gamma_{j}=(f_{j}=0)$, $1\leq j\leq r$. Let $\Pi\colon(M,D)\to(\mathbb{C}^{2},0)$ be the minimal reduction of singularities of $X$. Denote as $\widetilde{\mathcal{F}}_{X}$ and $\widetilde{\mathcal{F}}_{Y}$ the foliations given by the strict transforms of $\Pi^{*}(X)$ and $\Pi^{*}(Y)$, respectively.

Claim 3.3**.**

In the above situation we have

• (a).

$\mu(Y,0)\geq\mu(X,0)$.

• (b).

$Sing(\widetilde{\mathcal{F}}_{X})\cap D\subset Sing(\widetilde{\mathcal{F}}_{Y})\cap D$.

• (c).

For any $p\in Sing(\widetilde{\mathcal{F}}_{X})\cap D$ we have $\mu(\Pi^{*}(X),p)\leq\mu(\Pi^{*}(Y),p)$.

Proof. Since $X$ is G.C, a result of [3] says that the reduction of $X$ coincides with the reduction of the curve $\Gamma$, which in fact, coincides with the reduction of singularities of the foliation given by $d(f_{1}...f_{r})=0$. Note that $X$ has no other separatrices than that defined by the $f_{j^{\prime}s}$. Moreover, in [3] is also proved that:

• (i).

$\mu(X,0)=\mu(d(f_{1}...f_{r}),0)=\sum_{j=1}^{r}\mu(f_{j},0)-1$.

• (ii).

If $Z$ is any vector field such that $\Gamma_{1},...,\Gamma_{r}$ are separatrices of $Z$ then $\mu(Z,0)\geq\mu(X,0)$.

• (iii).

If $Z$ is as in (ii) and $D_{j}$ is any irreducible component of the exceptional divisor $D$, then the order of annulment of $\Pi^{*}(Z)$ along $D_{j}$ is $\geq$ the order of annulment of $\Pi^{*}(X)$ along $D_{j}$. In other words, if in local coordinates $(x,y)$ near some $z\in D_{j}$ we have $D_{j}=(x=0)$, $\Pi^{*}(X)=x^{k}\,\widetilde{X}$ and $\Pi^{*}{Z}=x^{\ell}\,\widetilde{Z}$, where $\widetilde{X}$ and $\widetilde{Z}$ have isolated singularities, then $k\leq\ell$. We will denote $\ell:=\mu(\Pi^{*}(Z),D_{j})$.

In particular, (ii) $\implies$ (a). Let us prove (b) and (c). Since the reduction of singularities of $X$ and $d(f_{1}...f_{r})=0$ coincide, if $p\in Sing(\widetilde{\mathcal{F}}_{X})\cap D$ then:

• I.

either $p$ is a corner of $D$: $p=D_{i}\cap D_{j}$, $i\neq j$,

• II.

or $p$ corresponds to the intersection of the strict transform of some of the curves $\Gamma_{j}$ with some irreducible component $D_{i}$ of $D$. In this case, $p\notin D_{\ell}$ if $\ell\neq i$.

In any case, I or II, $p$ must be also a singularity of the strict transform $\widetilde{\mathcal{F}}_{Y}$, because all the curves $\Gamma_{j}$, $1\leq j\leq r$, are $Y$-invariant. Since all singularities of $\widetilde{\mathcal{F}}_{X}$ are reduced, we have $\mu(\widetilde{\mathcal{F}}_{X},p)=1\leq\mu(\widetilde{\mathcal{F}}_{Y},p)$. Finally, in case I, $p=D_{i}\cap D_{j}$, we have

[TABLE]

[TABLE]

whereas in case II, we have

[TABLE]

Let us finish the proof of theorem 4. Let $p$ be a non-rational singularity of $\widetilde{\mathcal{F}}_{X}$. By claim 3.2, after a formal change of variables $\Phi$ and multiplication by a constant, we have

[TABLE]

where $p+q\geq 1$ and $\lambda\notin\mathbb{Q}$. Set also $\widehat{Y}:=\Phi^{*}(\Pi^{*}(Y))$. Note that $[X,Y]=0$ implies that $[\widehat{X},\widehat{Y}]=0$.

On the other hand, the vector field $Z:=q\,x\frac{\partial}{\partial x}-p\,y\frac{\partial}{\partial y}$ commutes with $\widehat{X}$: $[Z,\widehat{X}]=0$. Since $\widehat{X}\wedge Z\not\equiv 0$ we can write

[TABLE]

[TABLE]

$\phi$ and $\psi$ are (formal) meromorphic first integrals of $\widehat{X}$ and also of $x\frac{\partial}{\partial x}+\lambda\,y\frac{\partial}{\partial y}$. But, since $\lambda\notin\mathbb{Q}$, $\phi$ and $\psi$ must be constants, which implies $\phi=c\in\mathbb{C}$ and $\psi=0$ (because $\mu(\widehat{Y},p)>1$). Hence, $Y=c.\,X$ and $Y\in\mathbb{C}.\,X$. ∎

This motivates the following problem:

Problem 4**.**

Let $X\in\mathcal{X}_{2}$ be a non-dicritical G.C with an isolated singularity at $0\in\mathbb{C}^{2}$ and $DX(0)=0$. Assume also that $r(X)=2$. Is it true that $X$ has a non-constant holomorphic first integral?

When $DX(0)\neq 0$ then the answer is negative, as shows the following example:

Example 8**.**

Let

[TABLE]

where $\lambda\notin\mathbb{Q}$. Then $C(X)=\left<X,x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}\right>_{\mathbb{C}}$, so that $r(X)=d(X)=2$, but $\mathcal{I}(X)=\mathbb{C}$.**

3.4. Homogeneous and quasi-homogeneous vector fields.

Let $P=(p_{1},...,p_{n})\in\mathbb{Z}_{\geq 1}^{n}$ with $gcd(p_{1},...,p_{n})=1$. We say that $f\in\mathcal{O}_{n}$ is $P$ quasi-homogenous of degree $k\in\mathbb{N}$ if

[TABLE]

It is known that (10) implies that $f$ is a polynomial. We would like to observe the following: let $S$ be the linear vector field $S=\sum_{j=1}^{n}p_{j}\,z_{j}\,\frac{\partial}{\partial z_{j}}$. Then

[TABLE]

We will say also that $f$ is $S$ quasi-homogeneous. When $S=R=\sum_{j}z_{j}\,\frac{\partial}{\partial z_{j}}$, the radial vector field, then $R(f)=k.\,f$ $\iff$ $f$ is homogeneous of degree $k$.

By analogy, in the case of vector fields, we will say that $X$ is $S$ quasi-homogeneous if $[S,X]=k.\,X$ for some $k\in\mathbb{Z}$. For instance, if $S=R$ then $[R,X]=k.\,X$ $\iff$ the coefficients of $X$ are homogeneous polynomials of degree $k+1$.

Note also thar, if $S=\sum_{j}p_{j}\,z_{j}\frac{\partial}{\partial z_{j}}$ and $X=\sum_{j}X_{j}\frac{\partial}{\partial z_{j}}$ then $[S,X]=k\,X$ $\iff$ $X_{j}$ is $P$ quasi-homogeneous of degree $k+p_{j}$, $1\leq j\leq n$.

Remark 3.4**.**

Let $S=\sum_{j=1}^{n}p_{j}\,z_{j}\frac{\partial}{\partial z_{j}}$, $p_{j}\in\mathbb{N}$, $1\leq j\leq n$. Let $F_{k}:=\{f\in\mathcal{O}_{n}\,|\,S(f)=k.\,f\}$ and $E_{k}:\{X\in\mathcal{X}_{n}\,|\,[S,X]=k.\,X\}$. The following facts are well known:

If $f\in F_{k}$ (resp. $X\in E_{k}$) then $f$ is a polynomial (resp. $X$ is a polynomial vector field).

• 2.

$\mathcal{O}_{n}=\bigoplus_{k\geq 0}F_{k}$ and $\mathcal{X}_{n}=\bigoplus_{k\in\mathbb{Z}}E_{k}$. In particular, any $f\in\mathcal{O}_{n}$ (resp. any $X\in\mathcal{X}_{n}$) can be expressed as a convergent series $f=\sum_{k\geq 0}f_{k}$ (resp. $X=\sum_{k\in\mathbb{Z}}X_{k}$), where $f_{k}\in F_{k}$ (resp. $X_{k}\in E_{k}$), for all $k$. Analogously, any $\widehat{f}\in\widehat{\mathcal{O}}_{n}$ (resp. $\widehat{X}\in\widehat{\mathcal{X}}_{n}$) can be decomposed as a formal power series in $\bigoplus_{k\geq 0}F_{k}$ (resp. $\bigoplus_{k\in\mathbb{Z}}E_{k}$).

• 3.

If $f\in F_{k}$, $g\in F_{\ell}$ and $X\in E_{m}$ then $f.\,g\in F_{k+\ell}$, $f.\,X\in E_{k+m}$ and $X(f)\in F_{k+m}$.

• 4.

Let $f\in E_{k}\setminus\{0\}$, $k>0$, so that $f(0)=0$. Let $f=\Pi_{j=1}^{r}f_{j}^{m_{j}}$be the decomposition of $f$ into irreducible factors. Then we can assume that $f_{j}$ is $S$ quasi-homogeneous; $S(f_{j})=k_{j}.\,f_{j}$, $1\leq j\leq r$, where $k=\sum_{j}m_{j}\,k_{j}$.

• 5.

If $X\in E_{k}$ and $Y\in E_{\ell}$ then $[X,Y]\in E_{k+\ell}$

In the two dimensional case we have the following:

Proposition 9**.**

Let $S=p\,x\frac{\partial}{\partial x}+q\,y\frac{\partial}{\partial y}$ and $X=X_{1}\frac{\partial}{\partial x}+X_{2}\frac{\partial}{\partial y}\in\mathcal{X}_{2}$ with $[S,X]=k\,X$. Let $\omega:=i_{X}dx\wedge dy=X_{1}dy-X_{2}dx$. Assume that $S\wedge X=h.\,\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}$, where $h=p\,x.\,X_{2}-q.\,y\,X_{1}\not\equiv 0$. Then $h\in F_{k+tr(S)}$, $tr(S)=p+q$, and is an integrating factor of $\omega$: $d\left(\frac{\omega}{h}\right)=0$. Moreover, if $h=\Pi_{j=1}^{r}h_{j}^{k_{j}}$ is the decomposition of $h$ into $S$ quasi-homogeneous factors, then

• (a).

There exist $\lambda_{1},...,\lambda_{r}\in\mathbb{C}$ and $\varphi$, $S$ quasi-homogeneous, such that

[TABLE]

• (b).

The curves $\Gamma_{j}:=(h_{j}=0)$, $1\leq j\leq r$, are all separatrices of $X$ and $S$.

• (c).

If $X$ has other separatrices than the $\Gamma_{j`s}$ then $X$ is dicritical.

Proof. Set $\mu=\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}$. If $Y=Y_{1}\frac{\partial}{\partial x}+Y_{2}\frac{\partial}{\partial y}\in\mathcal{X}_{2}$ then $L_{Y}\mu=-\nabla Y.\,\mu$ where $\nabla Y=\frac{\partial Y_{1}}{\partial x}+\frac{\partial Y_{2}}{\partial y}$. In particular

[TABLE]

[TABLE]

The proof that $h$ is an integrating factor of $\omega$ and of (11) can be found in [4].

Let us prove that $(h=0)$ is $X$-invariant. It is enough to prove that $h|X(h)$ (see [4]). We have

[TABLE]

From the above relation and $h=\Pi_{j}h_{j}^{k_{j}}$ we get

[TABLE]

$\Gamma_{j}$ is $X$-invariant, $1\leq j\leq r$.

Let us prove (c). Let $t\in\mathbb{C}\mapsto S_{t}:=exp(t.\,S)$ be the flow of $S$. We assert that $S_{t}$ sends separatrices of $X$ onto separatrices of $X$.

In fact, relation $[S,X]=k\,X$ is equivalent to $S_{t}^{*}(X)=e^{k\,t}.\,X$. Let $f$ be an equation of a separatrix of $X$, so that $X(f)=g.\,f$. If $t\in\mathbb{C}$ is fixed, we have

[TABLE]

[TABLE]

$(f\circ S_{t}=0)$ is a separatrix of $X$.

If $X$ has a separatrix $\Gamma\notin\{\Gamma_{1},...,\Gamma_{r}\}$ then this separatrix cannot be $S_{t}$-invariant, because otherwise its equation would be contained in the equation $h=0$ (remember that $S\wedge X=h\,\mu$). In this case, the set $\{S_{t}(\Gamma)\,|\,t\in\mathbb{C}\}$ would contain a non-countable set of separatrices of $X$ and $X$ is dicritical. ∎Concerning the existence of non constant first integrals in the case of dimension two, we have the following:

Proposition 10**.**

Let $X,Y\in\mathcal{X}_{2}\setminus\{0\}$ be $S$ quasi-homogeneous, where $S=p\,x\frac{\partial}{\partial x}+q\,y\frac{\partial}{\partial y}$, $p,q\in\mathbb{Z}_{\geq 1}$. Assume that $Y\in C(X)$, $S\wedge X=h\,\mu$ and $X\wedge Y=f\,\mu$, where $h\not\equiv 0$. Then $f/h$ is a first integral of $X$. Moreover, if $X\in E_{k}$, $Y\in E_{\ell}$ and $0\neq\ell\neq k$ then $f/h$ is non-constant.

Proof. We assume $f\neq 0$. As we have seen $S\wedge X=h\,\mu$ implies that $X(h)=\nabla X.\,h$. On the other hand, $[X,Y]=0$ implies that

[TABLE]

[TABLE]

$f/h$ is a first integral of $X$.

Assume now that $X\in E_{k}$, $Y\in E_{\ell}$ and $0\neq\ell\neq k$. We have two possibilities:

1st. $f\neq 0$. We have

[TABLE]

Similarly, $h\in F_{k+tr(S)}$. In particular, $S(f/h)=\ell\,f/h\neq 0$ and so $f/h$ is a non constant first integral of $X$.

2nd. $f=0$. In this case, there exists a vector field $Z$, with isolated singularity at [math], and $\phi,\psi\in\mathcal{O}_{2}\setminus\{0\}$, such that $X=\phi.\,Z$ and $Y=\psi.\,Z$, so that $Y=g.\,X$, $g=\psi/\phi$. In particular, $[X,Y]=0$ implies that $X(g)=0$. Since $X\in E_{k}$ and $Y\in E_{\ell}$, $\ell\neq k$, the function $g$ cannot be constant. ∎

Theorem 5**.**

Let $X\in\mathcal{X}_{2}$ be $S=p\,x\frac{\partial}{\partial x}+q\,y\frac{\partial}{\partial y}$ quasi-homogeneous with $[S,X]=k.\,X$, $k>0$. Assume that $X$ has an isolated singularity at $0\in\mathbb{C}^{2}$ and is non-dicritical. If $C(X)\neq\mathbb{C}.\,X$ then $\mathcal{I}(X)=\mathbb{C}\{f\}$, where $f\neq 0$ is $S$ quasi-homogeneous and $C(X)=\mathbb{C}\{f\}.\,X$.

Remark 3.5**.**

The condition $[S,X]=k.\,X$, $k>0$, implies that $DX(0)$ is nilpotent.**

Proof. The hypothesis implies that there exists $Y\in C(X)$ such that $[X,Y]=0$ and $Y\notin\mathbb{C}.\,X$. Our purpose is to prove that $Y=\varphi(f).\,X$, where $f\in\mathcal{I}(X)$ and $\varphi\in\mathcal{O}_{1}$.

Let $Y=\sum_{j\in\mathbb{Z}}Y_{j}$ be the decomposition of $Y$ into $\mathcal{X}_{2}=\bigoplus_{j\in\mathbb{Z}}E_{j}$. Note that

[TABLE]

because $[X,Y_{\ell}]\in E_{k+\ell}$, $\forall\ell$. Let us see how $Y_{\ell}$ looks like. $1^{st}$ case: $\ell\notin\{0,k\}$. Let $Y_{\ell}\neq 0$ be such that $[X,Y_{\ell}]=0$. As we have seen in proposition 8, in this case $X$ has a non-constant meromorphic first integral, say $f$. Since $X$ is non-dicritical we can assume that $f$ is holomorphic and minimal, so that $\mathcal{I}(X)=\mathbb{C}\{f\}$: if $f$ was pure meromorphic then $X$ would have infinitely many separatrices and would be dicritical. We can assume that $f$ is quasi-homogeneous.

In fact, if $f=\sum_{j\geq 1}f_{j}$ is the decomposition of $f$ into $\mathcal{O}_{2}=\bigoplus_{j\geq 0}F_{j}$ then $X(f)=0$ implies that $X(f_{j})=0$ for all $j\geq 1$. Since $f\neq 0$, there is $r$ such that $f_{r}\neq 0$, so that $f=f_{r}\in F_{r}$ because $f$ is minimal. We are going to prove that $Y_{\ell}(f)=0$.

Assume that $Y_{\ell}(f)\neq 0$. Relation $[X,Y_{\ell}]=0$ implies that

[TABLE]

[TABLE]

Since $Y_{\ell}(f)\in F_{\ell+r}$ and $a_{j}.\,f^{j}\in F_{jr}$ we must have

[TABLE]

In this case, we have also $r\,m=\ell+r$ and $\ell=r(m-1)$ $\implies$ $m>1$.

Relation (9) implies also that

[TABLE]

On the other hand, $S(f)=r\,f$ implies that $f^{m-1}\,S(f)=r\,f^{m}$, and so

[TABLE]

Since [math] is an isolated singularity of $X$, we get

[TABLE]

Note that $g\in F_{\ell-k}$, because $Y_{\ell},\,g.\,X\in E_{\ell}$ and $X\in E_{k}$.

Relation (14) implies also:

[TABLE]

where in the last implication we have used proposition 8. From this and (13) we get

[TABLE]

However, since $g\in F_{\ell-k}$ and $Y_{\ell}\in E_{\ell}$ we get

[TABLE]

as we wished. Finally, since $f\in\mathcal{I}(X)\cap\mathcal{I}(Y_{\ell})$ we get $Y_{\ell}=h.\,X$, so that $X(h)=0$ and $h\in\mathbb{C}\{f\}$. In particular, we have $Y_{\ell}\in\mathcal{I}(X).\,X$. $2^{nd}$ case: $\ell=k$. In this case, we have shown in proposition 8 that if $X\wedge Y_{k}=\psi\,\mu$ and $S\wedge X=\phi\,\mu$, where $\mu=\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}$, then $X(\psi/\phi)=0$ (note that $\phi\neq 0$, because otherwise $X$ would be a multiple of $S$). Since $L_{S}\mu=-tr(S)\,\mu$ we get $\psi\in F_{2k+tr(s)}$ and $\phi\in F_{k+tr(s)}$ and so $S(\psi/\phi)=k.\,\psi/\phi$. We have two possibilities:

2.1. $\psi/\phi=c$, where $c\in\mathbb{C}$. In this case, necessarily $c=0$ and $\psi=0$, which implies $Y_{k}=g\,X$, where $g\in F_{0}$, so that $g$ is a constant and $Y_{k}\in\mathbb{C}.\,X$.

2.2. $\psi/\phi$ is not a constant. In this case, $X$ has a non constant first integral. Since $X$ is non-dicritical, it has a holomorphic minimal first integral, say $f$, where $f\in F_{k}$. As in the $1^{st}$ case, $Y_{k}(f)\in\mathbb{C}\{f\}$ and $Y_{k}(f)=a_{m}\,f^{m}$, where $f\in F_{r}$ and $k=r(m-1)$, so that $m>1$. As in the $1^{st}$ case, we can write

[TABLE]

but now $g\in F_{0}$ and so it is a constant. Hence,

[TABLE]

Therefore, in both cases we get $Y_{k}\in\mathbb{C}.\,X\subset\mathcal{I}(X).\,X$. $3^{rd}$ case: $\ell=0$. We assert that $Y_{0}=0$. In fact, as before, set $X\wedge Y_{0}=\psi\,\mu$ and $S\wedge X=\phi\,\mu$, $\phi\neq 0$, so that $X(\psi/\phi)=0$. Note that $\psi,\phi\in F_{k+tr(s)}$ and so $S(\psi/\phi)=0$. Therefore, if $\psi/\phi$ is not a constant then $X$ and $S$ would be colinear, a contradiction: $X$ would be dicritical. Hence, $X\wedge Y_{0}=c\,S\wedge X$ $\implies$ $X\wedge(Y_{0}+c\,S)=0$ $\implies$ $Y_{0}=-c\,S$, because $X$ has an isolated singularity at [math]. If $c\neq 0$ then $X\wedge S=0$, a contradiction. Therefore $Y_{0}=0$.

This finishes the proof of theorem 4. ∎

3.5. The dicritical case.

In this section we will assume that $X\in\mathcal{X}_{2}$ is dicritical. In this case, if $\Pi\colon(M,D)\to(\mathbb{C}^{2},0)$ is Seidenberg’s reduction of the singularities of $X$, then some of the irreducible components of the divisor $D$ are dicritical: non-invariant for the strict transform $\widetilde{\mathcal{F}}_{X}$ of the foliation $\mathcal{F}_{X}$, defined by $X$.

For instance, the foliation whoose leaves are the levels of a non-constant holomorphic function is non-dicritical, whereas the foliation whoose leaves are the levels of of pure meromorphic function is dicritical.

Example 9**.**

An interesting dicritical foliation (see [4]) is the one whoose leaves are the levels of the meromorphic function $(y^{2}+x^{3})/x^{2}$. The associated vector field is

[TABLE]

As a consequence of the next result we will prove that $C(X)=\mathbb{C}.\,X$.**

Example 10**.**

An example of dicritical vector field $X\in\mathcal{X}_{2}$ for which $\mu(X,0)=n\geq 2$ and $r(X)=2$ is

[TABLE]

If $n\geq 2$ then $C(X)=\mathbb{C}.\,x^{2}\frac{\partial}{\partial x}+\mathbb{C}.\,y^{2}\frac{\partial}{\partial y}$ and $r(X)=d(X)=2$. If $n=1$ then $X$ is the radial vector field and $d(X)=4$.

When $n\geq 2$ then $X$ has the meromorphic first integral $(y^{n-1}-x^{n-1})/x^{n-1}.\,y^{n-1}$.**

In the next result we consider a germ of dicritical vector field $X\in\mathcal{X}_{2}$ with an isolated singularity at $0\in\mathbb{C}^{2}$. Let

[TABLE]

be the blowing-up process of resolution of singularities of $X$. Denote by $\widetilde{\mathcal{F}}_{m}$ the strict transform of $\mathcal{F}_{X}$ by the composition $\widetilde{\Pi}_{m}:=\Pi_{1}\circ...\circ\Pi_{m}\colon(M_{m},E_{m})\to(\mathbb{C}^{2},0)$.

Definition 2**.**

Let $Z\in\mathcal{X}_{2}$ with first non-zero jet $Z_{\nu}=j^{\nu}(Z,0)$, $\nu\geq 1$, such that $Z_{\nu}\wedge R\equiv 0$. We will say that $Z$ has a purely radial singularity at [math] if $\nu=1$. In this case we have $Z=\alpha.\,R+h.o.t.$, where $R$ is the radial vector field and $\alpha\in\mathbb{C}^{*}$.

If $0\in\mathbb{C}^{2}$ is an isolated singularity of $Z$ and $\nu>1$ we will say that $Z$ has non purely radial singularity at [math] (briefly n.p.r.s). In this case, we have necessarily

[TABLE]

where $f$ is a homogeneous polynomial of degree $\nu-1$.

We will assume that at some step of the resolution, say $\Pi_{s}\colon(M_{s},E_{s})\to(M_{s-1},E_{s-1})$, $1\leq s\leq r$, we have the following:

• (1).

$\Pi_{s}$ is the blowing-up at a dicritical singularity $p_{o}\in E_{s-1}$ of $\widetilde{\mathcal{F}}_{s-1}$.

• (2).

If $s>1$ then $p_{o}$ belongs

• (2.a).

either to a unique divisor $\widetilde{D}\subset E_{s-1}$, which is a non-dicritical divisor of $\widetilde{\mathcal{F}}_{s-1}$,

• (2.b).

or to a corner $\widetilde{D_{1}}\cap\widetilde{D_{2}}$ of $E_{s-1}$, where both divisors are non-dicritical for $\widetilde{\mathcal{F}}_{s-1}$.

• (3).

The germ of $\widetilde{\mathcal{F}}_{s-1}$ is defined by a germ at $p_{o}$ of vector field $\widetilde{X}$ with a n.p.r.s at $p_{o}$. In other words, in some coordinates $(x,y)$ around $p_{o}$, the first non-zero jet $\widetilde{X}_{\nu}$ of $\widetilde{X}$ is of the form

[TABLE]

where $R=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ is the radial vector field and $f=f(x,y)$ is a non-constant homogeneous polynomial of degree $\nu-1\geq 1$.

Theorem 6**.**

If $X$ has an isolated singularity at $0\in\mathbb{C}^{2}$ and is as above then $C(X)=\mathbb{C}.\,X$.

Proof. The proof will be based in the following lemma:

Lemma 3.3**.**

Let $Z,W\in\mathcal{X}_{2}$ be such that $[W,Z]=0$ and $\Pi\colon(M,E)\to(\mathbb{C}^{2},0)$ be a blowing-up process, where $E$ is the exceptional divisor. Let $\widetilde{\mathcal{F}}_{Z}$ and $\widetilde{\mathcal{F}}_{W}$ denote the strict transforms of the foliations $\mathcal{F}_{Z}$ and $\mathcal{F}_{W}$ by $\Pi$, respectively. Suppose that the exceptional divisor $E$ has an irreducible component $D$ such that:

• (a).

$D$* is dicritical for $\widetilde{\mathcal{F}}_{Z}$.*

• (b).

$\Pi^{*}(W)|_{D}\equiv 0$.

Then $Z\wedge W\equiv 0$.

Proof. Since $D$ is dicritical for $\widetilde{\mathcal{F}}_{Z}$, if we fix a generic point $p\in D$ then there are local coordinates $(U,(x,y))$ around $p$ such that

• (i).

$x(p)=y(p)=0$ and $E\cap U=D\cap U=(y=0)$.

• (ii).

$\widetilde{\mathcal{F}}_{Z}$ is transverse to $D$ at $p$.

In particular, the germ of $\widetilde{\mathcal{F}}_{Z}$ is represented by a vector field $\widetilde{Z}$ transverse to $D$ at $p$. After a local change of variables and taking a smaller $U$ if necessary, we can assume that $\widetilde{Z}|_{U}=\frac{\partial}{\partial y}$. Since $\Pi|_{U\setminus D}$ is a biholomorphism, we must have $\Pi^{*}(Z)|_{U}=\varphi\frac{\partial}{\partial y}$, where $\varphi(q)\neq 0$ for all $q\in U\setminus D$.

Now, let

[TABLE]

where $A,B\in\mathcal{O}(U)$. From $[Z,W]=0$ we get

[TABLE]

A direct computation shows that the component of $\frac{\partial}{\partial x}$ of $[\Pi^{*}(Z),\Pi^{*}(W)]$ is $\pm\,\varphi.\,A_{y}$, which implies that $A_{y}=0$ and $A=A(x)$, so that $\Pi^{*}(W)=A(x)\frac{\partial}{\partial x}+B(x,y)\frac{\partial}{\partial y}$. Since $\Pi^{*}(W)|_{D}\equiv 0$ we get $A\equiv 0$, and so $\Pi^{*}(Z)\wedge\Pi^{*}(W)=0$ on $U$. But this implies $Z\wedge W\equiv 0$. ∎As an application we will prove theorem 6 when $p_{o}=0\in\mathbb{C}^{2}$, that is when $X=\sum_{j\geq\nu}X_{j}$, where $X_{\nu}=f.\,R$, $f$ homogeneous of degree $\nu-1\geq 1$.

Corollary 3.6**.**

If $X$ has a non purely radial singularity at $0\in\mathbb{C}^{2}$ and [math] is an isolated singularity of $X$ then $C(X)=\mathbb{C}.\,X$.

Proof. Fix $Y\in C(X)$. Let $\Pi\colon(M,D)\to(\mathbb{C}^{2},0)$ be the blow-up at $0\in\mathbb{C}^{2}$. Then $D$ is dicritical for $\Pi^{*}(X)$. We will divide the proof in two cases:

$1^{st}$* case:* $\mu(Y,0)\geq 2$. In this case $\Pi^{*}(Y)|_{D}\equiv 0$ and we can apply lemma 3.3 to show that $Y\wedge X\equiv 0$. Since $X$ has an isolated singularity at $0\in\mathbb{C}^{2}$ we get $Y=h.\,X$, where $h\in\mathcal{I}(X)$. But since $X$ is dicritical we get $h\in\mathbb{C}$ and $Y\in\mathbb{C}.\,X$.

$2^{nd}$* case:* $\mu(Y,0)=1$. We will see that this is impossible. In fact, in this case we have $X\wedge Y=h.\,\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}$, where $h(0)=0$ and $h\not\equiv 0$. We have seen that any irreducible component of $h$ is invariant by both vector fields $X$ and $Y$. Let $g$ be an irreducible component of $h$. By lemma 3.1 there exists $\lambda\in\mathbb{C}$ such that $Y-\lambda X=g.\,Z$, where $Z\in\mathcal{X}_{2}$. Since $\mu(Y,0)=1$ and $\mu(X,0)\geq 2$ we get

[TABLE]

$\mu(g,0)=1$ and $\mu(Z,0)=0$, because $g(0)=0$ and $g\not\equiv 0$. Since $Z(0)\neq 0$, after a change of variables we can suppose that $Z=\frac{\partial}{\partial x}$, so that $Y-\lambda.\,X=g.\,\frac{\partial}{\partial x}$, where $\mu(g,0)=1$. Let $g_{1}=a\,x+b\,y$ be the linear part of $g$ at $0\in\mathbb{C}^{2}$. We assert that $a=0$ and $b\neq 0$.

In fact, if $a\neq 0$ then for some $\lambda^{\prime}\neq\lambda$ the origin will be a saddle-node of $W:=Y-\lambda^{\prime}\,X$, but this is impossible: corollary 3.1 implies that $W$ is holomorphically normalizable and in § 2.3 it is proved that the pencil generated by $X$ and $Y$ must be equivalent to

[TABLE]

but then $\epsilon=1$ and $X=Z_{0}=x^{p}\,R$ and [math] is not an isolated singularity of $X$. Hence, $a=0$ and after dision by $b$ we can assume that $Y-\lambda\,X=y\frac{\partial}{\partial x}$. However, again by § 2.3, we must have $X\in\mathbb{C}\{y\}.\left<R,\frac{\partial}{\partial x}\right>$ which implies that $X$ cannot have an isolated singularity at $0\in\mathbb{C}^{2}$ with $X_{\nu}=f.\,R$, $deg(f)\geq 1$. ∎In the general case, the idea is similar. Recall the blowing-up process of the resolution of singularities of $X$ in (15) with $r\geq 1$ steps. In the $k^{th}$ step $\widetilde{\Pi}_{k}\colon(M_{k},E_{k})\to(\mathbb{C}^{2},0)$ we have called $\widetilde{\mathcal{F}}_{k}$ the foliation induced by the strict transform $\widetilde{\Pi}_{k}^{*}(X)$. From now on we will assume $r\geq 2$ and that the point $p_{o}$ which is a n.p.r.s of $\widetilde{\mathcal{F}}_{s-1}$ appears in the $(s-1)$-step, where $s\geq 2$. Fix $Y\in C(X)$. Given $1\leq m\leq r$ let us denote by $\widetilde{\mathcal{F}}_{m}^{Y}$ the strict transform of the foliation defined by $Y^{*}:=\widetilde{\Pi}_{m}^{*}(Y)$. In order to apply lemma 3.3, we have to prove that it is possible to find a n.p.r.s $p_{o}\in Sing(\widetilde{\mathcal{F}}_{s-1})$ with the property that $\mu(Y^{*},p_{o})\geq 2$. In this case, if we blow-up at $p_{o}$, $\Pi_{s}\colon(M_{s},D)\to(M_{s-1},p_{o})$ then $D$ will be dicritical for $\widetilde{\mathcal{F}}_{s}$ and $\Pi_{s}(Y^{*})|_{D}\equiv 0$, so that we can apply lemma 3.3. In order to simplify the proof we will assume the following about the blowing-up process:

• •

When we pass from the $(m-1)^{th}$ step to the $m^{th}$ step by $\Pi_{m}\colon(M_{m},E_{m})\to(M_{m-1},E_{m-1})$ we don’t blow-up at a point $q\in Sing(\widetilde{\mathcal{F}}_{m-1})$ if it is a n.p.r.s or if it is purely radial. In other words, the blow-up $\Pi_{m}$ is done at a point $p\in Sing(\widetilde{\mathcal{F}}_{m-1})$ only if it is not a simple singularity and if $\Pi^{-1}(p)=D\subset E_{m}$ is a non-dicritical divisor for $\widetilde{\mathcal{F}}_{m}$.

Although the final foliation in this process has non-simple singularities, if $\widetilde{\Pi}_{n}=\Pi_{1}\circ\Pi_{2}\circ...\circ\Pi_{n}\colon(M_{n},E_{n})\to(\mathbb{C}^{2},0)$ is the final step, with this convention, then $\widetilde{\mathcal{F}}_{n}$ satisfies the following:

All irreducible components of $E_{n}$ are non-dicritical for $\widetilde{\mathcal{F}}_{n}$.

• 2.

A non-simple singularity $p\in Sing(\widetilde{\mathcal{F}}_{n})$ is, either purely radial, or n.p.r.s.

• 3.

$\widetilde{\mathcal{F}}_{n}$ has at least one n.p.r.s, say $p_{o}\in E_{n}$.

If we suppose that $0\in\mathbb{C}^{2}$ is not n.p.r.s then:

Claim 3.4**.**

We have two possibilities for $X$:

• $1^{st}$.

$\mu(X,0)\geq 2$. In this case, in any step $\widetilde{\Pi}_{m}\colon(M_{m},E_{m})\to(\mathbb{C}^{2},0)$ of the blowing-up process, in all non-dicritical divisors $D\subset E_{m}$ we have $\widetilde{\Pi}^{*}(X)|_{D}\equiv 0$.

• $2^{nd}$.

$\mu(X,0)=1$* and the linear part of $X$ at [math] is nilpotent. Moreover, if $D$ is a non-dicritical irreducible component of $E_{m}$ which is not the strict transform of $E_{1}$ by*

[TABLE]

then $\widetilde{\Pi}_{m}^{*}(X)|_{D}\equiv 0$.

Proof. The $1^{st}$ assertion is proved applying inductively lemma 3.2 in the process. We leave the details for the reader.

On the other hand, if $\mu(X,0)=1$, the first blow-up is not dicritical for $X$ and $\lambda_{1}$ and $\lambda_{2}$ are the eigenvalues of $DX(0)$ then:

• (i).

If $\lambda_{1},\lambda_{2}\neq 0$ then in the process of resolution of $X$ there is no n.p.r.s.

• (ii).

If $\lambda_{1}\neq 0$ and $\lambda_{2}=0$ then [math] is a saddle-node for $X$, which is not possible with our hypothesis.

Therefore, $DX(0)$ is nilpotent and we can assume that $DX(0)=y\frac{\partial}{\partial x}$. In this case, in the first blow-up $\Pi_{1}\colon(M_{1},E_{1})\to(\mathbb{C}^{2},0)$ then $\Pi_{1}^{*}(X)$ has only one singularity at $E_{1}\simeq\mathbb{P}^{1}$ corresponding to the direction $p=(y=0)$. This singularity is of algebraic multiplicity $\mu(\Pi_{1}^{*}(X),p)=2$. Therefore, the $2^{nd}$ assertion is also consequence of lemma 3.2. ∎Now, given $Y\in C(X)$ let $\widetilde{\mathcal{F}}_{n}^{Y}$ be as before, the strict transform of the foliation associated to $\widetilde{\Pi}_{n}^{*}(Y)$. Suppose, by contradiction that $X\wedge Y\not\equiv 0$.

Note first that all irreducible components of $E_{n}$ are non-dicritical for $\widetilde{\mathcal{F}}_{n}^{Y}$. Let us prove this fact.

Suppose first that $\mu(X,0)\geq 2$. In this case, if $D$ is an irreducible component of $E_{n}$ then $\widetilde{\Pi}_{n}^{*}(X)|_{D}\equiv 0$ by claim 3.4. On the other hand, if $D$ was dicritical for $\widetilde{\mathcal{F}}_{n}^{Y}$ then $X\wedge Y\equiv 0$ by lemma 3.3, a contradiction.

Suppose now that $\mu(X,0)=1$ and $DX(0)=y\frac{\partial}{\partial x}$. Since $\widetilde{\Pi}_{n}^{*}(X)|_{D}\equiv 0$ except for $D_{1}$, the strict transform of $E_{1}$, we conclude from lemma 3.3 that the unique irreducible component that could be dicritical for $\widetilde{\mathcal{F}}_{n}^{Y}$ is $D_{1}$. On the other hand, if $D_{1}$ was dicritical for $\widetilde{\mathcal{F}}_{n}^{Y}$ then necessarily $E_{1}$ is dicritical for the first blow-up of $Y$. This implies that the first non-zero jet of $Y$ is of the form $Y_{k}=g.\,R$, where $g$ is a homogeneous polynomial. The polynomial $g$ is necessarily non constant, for otherwise by Poincaré linearization theorem we can assume that $Y=R$, which implies that $X$ is a linear vector field contradicting the hypothesis. Now, with an argument similar to the $2^{nd}$ case in the argument of the proof of corollary 3.6 it can be proved that this is impossible. We leave the details for the reader. ∎Given an irreducible component $D\subset E_{m}$ we denote as $\mu(Y,D)$ (resp. $\mu(X,D)$) the order of annihilation of $\widetilde{\Pi}_{m}^{*}(Y)$ (resp. $\widetilde{\Pi}_{m}^{*}(X)$) along $D$. In other words, given $p\in D$ and a local coordinate system $(U,(x,y))$ such that $D\cap U=(y=0)$ then $\mu(Y,D)=k$ if $\widetilde{\Pi}_{m}^{*}(Y)=y^{k}\,\widetilde{Y}$ where $y\nmid\,\widetilde{Y}$. We have the following:

Claim 3.5**.**

Let $p\in D\subset E_{m}$. Assume that:

• (a).

$p$* is not a singularity of $\widetilde{\mathcal{F}}_{m}^{Y}$.*

• (b).

$\mu(Y,D)=k\geq 0$.

Then:

• (1).

If $k=0$ then $p$ is not a singularity of $\widetilde{\mathcal{F}}_{m}$.

• (2).

If $k\geq 1$ and $p\in Sing(\widetilde{\mathcal{F}}_{m})$ then $p$ is a non-degenerate singularity of $\widetilde{\mathcal{F}}_{m}$. Moreover, the germ of $\widetilde{\mathcal{F}}_{m}$ at $p$ is equivalent to the Poincaré-Dulac normal form $(k\,x+\alpha.\,y^{k})\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$.

Proof. Since $D$ is non-dicritical for $\widetilde{\mathcal{F}}_{m}^{Y}$ and $p\notin Sing(\widetilde{\mathcal{F}}_{m}^{Y})$ we can find a local coordinate system $(U,(x,y))$ around $p$ such that $D\cap U=(y=0)$ and $\widetilde{\mathcal{F}}_{m}^{Y}$ is defined by $\widetilde{Y}=\frac{\partial}{\partial x}$. Since $\mu(Y,D)=k$, we can assume that $\widetilde{\Pi}_{m}^{*}(Y)|_{U}=y^{k}.\,\frac{\partial}{\partial x}$. Let $\Pi^{*}(X)|_{U}=y^{\ell}\,\widetilde{X}$, where $\ell\geq 0$, $\widetilde{X}=A(x,y)\frac{\partial}{\partial x}+B(x,y)\frac{\partial}{\partial y}$ and $y\nmid\widetilde{X}$. From $[\widetilde{\Pi}_{m}^{*}(X)\,,\,\widetilde{\Pi}_{m}^{*}(Y)]=0$ we get

[TABLE]

[TABLE]

As the reader can check directly, the last relation implies that $B_{x}=0$ and $y.\,A_{x}=k.\,B$, so that

[TABLE]

If $k=0$ then, either $A(0)\neq 0$, or $B(0)\neq 0$, because $y\nmid\widetilde{X}$ and so $\widetilde{X}(p)\neq 0$.

If $k\geq 1$ and $p\in Sing(\widetilde{\mathcal{F}}_{m})$ then $a(0)=0$ and $b(0)\neq 0$, for otherwise $y|\widetilde{X}$. In particular, the eigenvalues of $D\widetilde{X}(0)$ are $b(0)$ and $k\,b(0)$. This implies the last assertion of (2). ∎

Corollary 3.7**.**

If in some step of the resolution of $X$, say $\Pi_{m}\colon(M_{m},E_{m})\to(M_{m-1},E_{m-1})$, where $1\leq m<n$, we explode at a point $p$ that is not a singularity of $\widetilde{\mathcal{F}}_{m-1}^{Y}$ then $\Pi_{m}\circ...\circ\Pi_{n}(p_{o})\neq p$, where $p_{o}\in E_{n}$ is the n.p.r.s singularity of $\widetilde{\mathcal{F}}_{n}$.

Proof. The proof is by contradiction: if not, then let $m$ be the smallest step in which we explode at a point $p\in E_{m-1}$ which is not a singularity of $\widetilde{\mathcal{F}}_{m-1}^{Y}$. Then, since $p$ is a singularity of $\widetilde{\mathcal{F}}_{m-1}$, by claim 3.5 the germ of $\widetilde{\mathcal{F}}_{m-1}$ at $p$ is represented by a vector field $\widetilde{X}$ which is equivalent to the Poincaré-Dulac normal form

[TABLE]

But when we continue the process after the resolution of $\widetilde{X}$ we don’t obtain any n.p.r.s singularity, a contradiction. ∎

Let us see how looks like $\widetilde{\Pi}_{m}^{*}(Y)$ in a neighborhood of a point $p\in E_{m}$, $1\leq m\leq n$. Denote as $\widetilde{\mathcal{F}}_{m}^{Y}$ the strict transform of $\mathcal{F}_{Y}$ by $\widetilde{\Pi}_{m}$. For the first blow-up we have the following:

Claim 3.6**.**

• (a).

If $\mu(X,0)\geq 2$ and $\mu(Y,0)=1$ then $DY(0)$ is nilpotent.

• (b).

If $\mu(X,0)=1$ and $DX(0)$ is nilpotent then $DY(0)$ is also nilpotent and $DY(0)\wedge DX(0)=0$.

Proof. Let $DY(0)=S+N$, where $S=\lambda_{1}\,x\frac{\partial}{\partial x}+\lambda_{2}\,y\frac{\partial}{\partial y}$ is semi-simple and $N$ is nilpotent and $[S,N]=0$. We assert that $S=0$.

In fact, suppose by contradiction that $S\neq 0$. Let $\mu(X,0)=k$ and $X_{k}=j^{k}(X,0)$ be the first non-zero jet of $X$. Note that

[TABLE]

Suppose first that $k\geq 2$. In this case, we must have $\lambda_{1},\lambda_{2}\neq 0$, for otherwise for $\alpha\in\mathbb{C}^{*}$ we have $Z:=Y+\alpha.\,X\in C(X)$ has a saddle-node at $0\in\mathbb{C}^{2}$, which is not possible with our hypothesis. On the other hand, if $\lambda_{1},\lambda_{2}\neq 0$, then $[S,X_{k}]=0$ implies that $S$ has a resonance and necessarily, after multiplying by a constant is equivalent to either $S=x\frac{\partial}{\partial x}+n\,y\frac{\partial}{\partial y}$, $n\in\mathbb{N}$, or to $S=m\,x\frac{\partial}{\partial x}-n\,y\frac{\partial}{\partial y}$, $m,n\in\mathbb{N}$.

Let $p_{o}\in E_{n}$ be the n.p.r.s singularity of $\widetilde{\mathcal{F}}_{n}$ and consider the sequence of images of $p_{o}$

[TABLE]

By corollary 3.7 all points in the sequence are singularities of the strict transform of $\mathcal{F}_{Y}$. Given $1\leq j\leq n$ let $D_{j}:=\Pi^{-1}_{j}(p_{n-j+1})$ be the irreducible component of $E_{j}$ obtained in the blowing-up $\Pi_{j}$. We assert that $\mu(\widetilde{\Pi}_{j}^{*}(Y),p_{n-j})=1$ and $\mu(\,\widetilde{\Pi}^{*}_{j}(Y),D_{j})=0$, $1\leq j\leq n$.

The above assertion is consequence of the following:

A. When we blow-up a non-degenerated and non radial singularity of a germ at $0\in\mathbb{C}^{2}$ of vector field $Z$, say by $\Pi\colon(\widetilde{\mathbb{C}}^{2},D)\to(\mathbb{C}^{2},0)$, $\mu(\Pi^{*}(Z),D)=0$ and we have two possibilites for the singularities of $\Pi^{*}(Z)$:

• A.1.

$\Pi^{*}(Z)$ has two non-degenerated singularities in the divisor $D$.

• A.2.

$\Pi^{*}(Z)$ has only one singularity in $D$, which is saddle-node. This happens only when $DZ(0)$ is equivalent to $R+y\frac{\partial}{\partial x}$.

B. When we blow-up at a saddle-node then it appears two singularities at the divisor, one non-degenerated and the other a saddle-node.

If we apply A and B inductivelly we obtain the assertions. Moreover, at the end of the process the vector field $Y^{*}:=\widetilde{\Pi}_{n}^{*}(Y)$ has, either a non-degenerated singularity, or a saddle-node at $p_{o}$. However this is not possible because $\widetilde{\Pi}_{n}^{*}(X)$ has a n.p.r.s singularity at $p_{o}$, as the reader can check. Suppose now that $k=1$. In this case $DX(0)$ is nilpotent and we can assume that $DX(0)=y\frac{\partial}{\partial x}$. Since $DY(0)$ and $DX(0)$ commute we can write $DY(0)=a\,R+b\,DX(0)$, $R$ the radial vector field. We assert that $a=0$. In fact, if $a\neq 0$ then by Poincaré’s linearization theorem we can assume that $Y=a.\,R+b\,DX(0)$. However, in this case $X$ cannot satisfy hypothesis (3) (see § 2.3). Therefore, $DY(0)$ is nilpotent and $DX(0)\wedge DY(0)=0$. ∎

Now, let $p_{o}\in E_{n}$ be a n.p.r.s: the germ at $p_{o}$ of $\widetilde{\mathcal{F}}_{n}$ is defined by $\widetilde{X}$, where the first non-zero jet of $\widetilde{X}$ is $\widetilde{X}_{k}=f.\,R$, $f$ homogeneous of degree $\nu-1\geq 1$. In order to apply lemma 3.3 it is suficient to prove that $\mu(\widetilde{\Pi}_{n}^{*}(Y),p_{o})\geq 2$. Note that, in principle, $p_{o}$ could be a pole of $\widetilde{\Pi}_{n}^{*}(Y)$, if at some steps of the process we had blow-up at points that are not singularities of strict transform of $\mathcal{F}_{Y}$. However, this doesn’t happens by corollary 3.7. Proof that $\mu(\widetilde{\Pi}_{n}^{*}(Y),p_{o})\geq 2$. Suppose by contradition that $\mu(\widetilde{\Pi}_{n}^{*}(Y),p_{o})=1$. Recall that $p_{o}\in Sing(\widetilde{\mathcal{F}}_{n}^{Y})$ by corollary 3.7. This implies that $p_{o}$ is an isolated singularity of $\widetilde{\Pi}_{n}^{*}(Y):=\widetilde{Y}$, so that $\widetilde{Y}$ defines the germ at $p_{o}$ of $\widetilde{\mathcal{F}}_{n}^{Y}$.

Choose coordinates $(U,(x,y))$ such that $x(p_{o})=y(p_{o})=0$ and

• (iii).

$p_{o}\in D$, where $D$ is an irreducible component of $E_{n}$. Moreover, $D\cap U=(y=0)$ and $\mu(\widetilde{\Pi}_{n}^{*}(X),D)=k\geq 0$.

• (iv).

The germ of $\widetilde{\mathcal{F}}_{n}$ is defined by $\widetilde{X}\in\mathcal{X}_{2}$ where $j^{\nu}(\widetilde{X})=X_{\nu}=f.\,R$, where $f$ is homogeneous of degree $\nu-1$. Moreover, $\widetilde{\Pi}_{n}^{*}(X)=y^{k}\,\widetilde{X}$.

• (v).

$\widetilde{Y}=A(x,y)\frac{\partial}{\partial x}+B(x,y)\frac{\partial}{\partial y}$, where $D\widetilde{Y}(0)\neq 0$. We have seen before that $D\widetilde{Y}(0)$ is nilpotent, so that we can suppose that $D\widetilde{Y}(0)=y\frac{\partial}{\partial x}$.

Since $[\widetilde{\Pi}_{n}^{*}(X),\widetilde{Y}]=0$, if we set $\widetilde{\Pi}_{n}^{*}(X)\wedge\widetilde{Y}=g.\,\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}$, then any irreducible component of $g$ is $\widetilde{Y}$ and $\widetilde{\Pi}_{n}^{*}(X)$ invariant. Let $h$ be an irreducible component of $g$. Then there exists $\lambda\in\mathbb{C}$ such that $h|\widetilde{Y}-\lambda.\,\widetilde{\Pi}_{n}^{*}(X)$:

[TABLE]

From the above relation we get

[TABLE]

As in the proof of corollary 3.6 we can assume that $Z=\frac{\partial}{\partial x}$ (recall that $D$ is non-dicritical for $\widetilde{Y}$) and $h=y$. The vector field $\widetilde{\Pi}_{n}^{*}(X)$ in the new coordinate system can still be written as $y^{k}\widetilde{X}$ where $0=p_{o}$ is an isolated singularity of $\widetilde{X}$, as the reader can check. In particular,

[TABLE]

[TABLE]

but then $p_{o}$ cannot be a singularity of the type n.p.r.s. This finishes the proof of theorem 6. ∎We have the following consequence of theorem 6:

Corollary 3.8**.**

Let $X\in\mathcal{X}_{2}$ be a dicritical vector field with an isolated singularity at $0\in\mathbb{C}^{2}$ such that $r(X)=2$. Let $\Pi\colon(M,E)\to(\mathbb{C}^{2},0)$ be the minimal resolution of the singularities of $X$ and $D_{i}\subset E$ be a dicritical divisor of $E$. Then $D_{i}^{2}=-1$ (the self intersection number) and $D_{i}$ is obtained by the blowing-up of a purely radial singularity on a non-dicritical divisor of the previous step of the resolution.

Another interesting result is the following:

Proposition 11**.**

Let $X\in\mathcal{X}_{2}$ be a dicritical vector field with an isolated singularity at $0\in\mathbb{C}^{2}$. Asume that $DX(0)$ is nilpotent. If $r(X)=2$ then $d(X)=2$.

Proof. As in the proof of theorem 6, consider the blowing-up process $\widetilde{\Pi}_{r}=\Pi_{1}\circ...\circ\Pi_{r}\colon(M_{r},E_{r})\to(\mathbb{C}^{2},0)$ as in 15. By theorem 6 during the process it never appears a n.p.r.s singularity. As a consequence, there is at least a dicritical exceptional divisor $D\subset E_{r}$ such that $D^{2}=-1$, which was obtained by blowing-up at a purely radial singularity on a non-dicritical divisor of a previous step. Without lost of generality, we will assume that $D=\Pi_{r}^{-1}(p_{o})$, where $p_{o}\in D^{\prime}\subset E_{n}$, $n=r-1$, is the purely radial singularity, and $D^{\prime}$ is the irreducible component of $E_{n}$ that contains $p_{o}$. Given $1\leq m\leq n$, set as before $\widetilde{\Pi}_{m}=\Pi_{1}\circ...\circ\Pi_{m}\colon(M_{m},E_{m})\to(\mathbb{C}^{2},0)$ and let $\widetilde{\mathcal{F}}_{m}$ be the strict transform of $\mathcal{F}_{X}$ at the level $m$: $\widetilde{\Pi}_{m}=\widetilde{\Pi}_{m}^{*}(\mathcal{F}_{X})$.

Remark 3.6**.**

If $D^{\prime}\subset E_{n}$ is the irreducible component that contains $p_{o}$ then $\mu(\widetilde{\Pi}_{n}^{*}(X),D^{\prime})\geq 1$. This is a direct consequence of claim 3.4.**

Claim 3.7**.**

There are coordinates $(U,(x,y))$ around $p_{o}$ such that:

$D^{\prime}\cap U=(y=0)$* and $x(p_{o})=y(p_{o})=0$.*

• 2.

$\widetilde{\Pi}_{n}^{*}(X)=y^{k}.\,R$, where $k=\mu(\widetilde{\Pi}_{n}^{*}(X),D^{\prime})\geq 1$ and $R$ is the radial vector field.

Proof. Let $(U^{\prime},(u,v))$ be local coordinates such that $u(p_{o})=v(p_{o})=0$ and $D^{\prime}\cap U^{\prime}=(v=0)$. Since $\mu(\widetilde{\Pi}_{n}^{*}(X),D^{\prime})=k$ we can write $\widetilde{\Pi}_{n}^{*}(X)=v^{k}.\,\widetilde{X}$, where $v\nmid\widetilde{X}$. By assumption $D\widetilde{X}(0)=\alpha\,R$, where $\alpha\neq 0$, so that by Poincaré’s linearization theorem, after a change of variables, we can assume that $\widetilde{X}=\alpha\,R$, $D^{\prime}\cap U^{\prime}=(v=0)$ and $\widetilde{\Pi}_{n}^{*}(X)=\phi.\,v^{k}.\,R$, where $\phi(0)\neq 0$. If we consider a change of variables of the form $h(u,v)=(\rho.\,u,\rho.\,v)=(x,y)$, where $\rho(0)\neq 0$, then

[TABLE]

Now, we use the fact that the differential equation $R(\rho)+\rho=\phi^{-1}\,\rho^{k+1}$ has a solution $\rho$ such that $\rho^{k}(0)=\phi(0)$. We leave the details for the reader. ∎Now, let $Y\in C(X)$ be such that $X\wedge Y\not\equiv 0$. We can assume that [math] is an isolated singularity of $Y$: if not then instead of $Y$ take $Y^{\prime}=Y+\alpha.\,X$, where $\alpha\neq 0$.

Denote as $\widetilde{\mathcal{F}}_{m}^{Y}$ the strict transform of $\widetilde{\Pi}_{m}^{*}(\mathcal{F}_{Y})$ on $M_{m}$, where $\widetilde{\mathcal{F}}_{0}=\mathcal{F}_{Y}$. As before, consider the sequence of images of $p_{o}$:

[TABLE]

Note that $p_{n-m}\in E_{m}$ if $1\leq m\leq n$.

Define $S(Y)=\{m\,|\,0\leq m\leq n$ and $p_{n-m}\in Sing(\widetilde{\mathcal{F}}_{m}^{Y})\}$.

Claim 3.8**.**

There exists $0<\ell<n$ such that $S(Y)=\{k\,|\,0\leq k\leq\ell\}$. Moroever $\mu(\widetilde{\Pi}_{n}^{*}(Y),D)=1$. In other words $p_{o}\notin Sing(\widetilde{\mathcal{F}}_{n}^{Y})$ and $\widetilde{\Pi}_{n}^{*}(Y)=y.\,\widetilde{Y}$, where $\widetilde{Y}(p_{o})\neq 0$ and $(y=0)$ is a local equation on $D^{\prime}$.

Proof. Of course $0=p_{n}\in Sing(\mathcal{F}_{Y})$. We will prove at the end that $S(Y)\neq\{0,...,n\}$. Assuming that $S(Y)\neq\{0,...,n\}$, there exists $0\leq m<n$ such that $0,...,m-1\in S(Y)$, but $m\notin S(Y)$.

Let $\widetilde{Y}$ and $\widetilde{X}$ be germs at $p_{n-m}$ of vector fields representing $\widetilde{\mathcal{F}}_{m}^{Y}$ and $\widetilde{\mathcal{F}}_{m}$, respectively. By claim 3.5, there are local coordinates $(U,(u,v))$ around $p_{n-m}$ and $\ell\geq 1$ such that $u(p_{n-m})=v(p_{n-m})=0$ and

• (i).

$E_{m}\cap U=(v=0)$.

• (ii).

$\widetilde{Y}=\frac{\partial}{\partial u}$ and $\widetilde{\Pi}_{m}^{*}(Y)=v^{\ell}.\,\widetilde{Y}$.

• (iii).

$\widetilde{X}=(\ell.\,u+\alpha\,v^{\ell})\frac{\partial}{\partial u}+v\,\frac{\partial}{\partial v}$.

Note that $\alpha=0$ because otherwise the singularity $p_{n-m}$ would not be dicritical. Following the resolution of $\ell.\,u\,\frac{\partial}{\partial u}+v\,\frac{\partial}{\partial v}$, we see that the radial singularity $p_{o}$ appears after $\ell-1$ blowing-ups and $m=n-\ell+1$. Moreover, the composition $\Pi:=\Pi_{n-\ell+2}\circ...\circ\Pi_{n}\colon(M_{n},p_{o})\to(M_{n-\ell+1},p_{n-m})$ in the chart that appears the radial singularity $p_{o}$ is of the form $\Pi(x,y)=(x.\,y^{\ell-1},y)=(u,v)$ with inverse $\Pi^{-1}(u,v)=(u/v^{\ell-1},v)=(x,y)$. It follows that

[TABLE]

This proves (b). It remains to prove that $S(Y)\neq\{0,...,n\}$.

Suppose by contradiction that $S(Y)\neq\{0,...,n\}$. In this case, all points in the sequence $p_{0},...,p_{n}$ are singularities of the strict transform of $\mathcal{F}_{Y}$. Note that $DY(0)$ is nilpotent. The proof of this fact is similar to the proof of claim 3.6. We leave the details for the reader.

Since all points of the sequence are singularities of the strict transform of $\mathcal{F}_{Y}$, when we apply lemma 3.2 inductively we find that $\mu(\widetilde{\Pi}_{m}^{*}(Y),p_{n-m})\geq 2$ if $1\leq m\leq n$. In particular, $\mu(\widetilde{\Pi}_{n}^{*}(Y),p_{o})\geq 2$. This implies that when we apply the blowing-up $\Pi_{n+1}\colon(M_{r},D)\to(M_{n},p_{o})$ then we have $\Pi_{n+1}^{*}\widetilde{\Pi}_{n}^{*}(Y)|_{D}\equiv 0$, which is impossible by lemma 3.3. ∎Let us finish the proof of proposition 11. Let $(U,(x,y))$ be a coordinate system around $p_{o}$ as in claim 3.7:

• (I).

$x(p_{o})=y(p_{o})=0$ and $D^{\prime}\cap U=(y=0)$.

• (II).

$\widetilde{\Pi}_{n}^{*}(X)=y^{k}.\,R$.

We assert that

[TABLE]

In fact, let $Z\in C(y^{k}\,R)$ and write $Z=\sum_{j\geq 0}Z_{j}$ where $Z_{j}$ is homogeneous of degre $j\geq 0$. From $[Z,y^{k}.\,R]=0$ we get

[TABLE]

[TABLE]

From (17) we get $Z_{0}=0$ and $Z_{j}\wedge R=0$ if $j\geq 2$. In particular, we can write $Z=Z_{1}+h.\,R$, where $h\in\mathcal{O}_{2}$ and $h(0)=0$. Note that the term of degree $2$ of the right side of (17) vanishes and this implies that $Z_{1}(y)=0$ so that $Z_{1}=(\alpha\,x+\beta\,y)\frac{\partial}{\partial x}$ and $[Z_{1},y^{k}\,R]=0$. It follows that

[TABLE]

From the above we can assume that $\Pi^{*}_{n}(Y)=y\,\frac{\partial}{\partial x}+c\,y^{k}\,R$, where $c\in\mathbb{C}$. Let $W\in C(X)$ with $W\wedge X\neq 0$. Since $\widetilde{\Pi}_{n}^{*}(W)|_{D^{\prime}}\equiv 0$, by the above argument we must have

[TABLE]

where $g$ is homogeneous of degree $k$. Since $W-\alpha\,Y\in C(X)$ and $\widetilde{\Pi}_{n}^{*}(W-\alpha\,Y)\wedge R\equiv 0$ we obtain $(W-\alpha\,Y)\wedge X\equiv 0$. Therefore $W=\alpha\,Y+\beta\,X$, $\beta\in\mathbb{C}$. This finishes the proof of proposition 11. ∎

Example 11**.**

An example satisfying proposition 11 is $X=y\frac{\partial}{\partial x}+x^{n}\,R$, $R$ the radial vector field. Note that $X$ is dicritical because admits the meromorphic first integral $\frac{1}{y^{n}}-\frac{n}{n+1}\left(\frac{x}{y}\right)^{n+1}$. In this case we have $C(X)=\left<X,y^{n}\,R\right>$.**

As a consequence of proposition 11, we have the following:

Corollary 3.9**.**

Let $X\in\mathcal{X}_{2}$ be a dicritical germ of vector field with an isolated singularity at $0\in\mathbb{C}^{2}$. Then $d(X)\leq 4$. Moreover:

• (a).

If $d(X)=3$ then $X$ is equivalent, modulo a multiplicative constant, to $x\frac{\partial}{\partial x}+n\,y\frac{\partial}{\partial y}$ where $n\in\mathbb{N}\setminus\{1\}$.

• (b).

If $d(X)=4$ then $X$ is equivalent, modulo a multiplicative constant, to the radial vector field.

Proof. By proposition 11 if $r(X)=2$ and $d(X)>2$ then $DX(0)$ cannot be nilpotent. Since $X$ is dicritical and cannot have a saddle-node at the origin $0\in\mathbb{C}^{2}$, we obtain that the semi-simple part of $DX(0)$, after multiplication by a constant, is of the form $m\,x\,\frac{\partial}{\partial x}+n\,y\,\frac{\partial}{\partial y}$, where $m,n\in\mathbb{N}$ and $gcd(m,n)=1$. In fact, from Poincaré’s linearization and Poincaré-Dulac theorems we can assume that, after multiplication by a constant, that $X$ is conjugated to one of the following vector fields:

$m\,x\frac{\partial}{\partial x}+n\,y\frac{\partial}{\partial y}$, where $1<m<n$. In this case we have $d(X)=2$ (see § 2.3).

• 2.

$x\frac{\partial}{\partial y}+n\,y\frac{\partial}{\partial y}$, $n>1$. In this case we have $d(X)=3$.

• 3.

$X=R$, the radial vector field. In this case we have $d(X)=4$.

Therefore, the corollary is a direct consequence of the examples in § 2.3. ∎Another consequence of theorem 6 and of proposition 11 is the following:

Corollary 3.10**.**

Let $X\in\mathcal{X}_{2}$ with an isolated singularity at $0\in\mathbb{C}^{2}$. If $d(X)>4$ then $X$ has a non-constant holomorphic first integral. In particular $d(X)=\infty$.

Proof. Note first that $X$ cannot be dicritical by corollary 3.9, because $d(X)>4$. Moreover, $d(X)>2\geq r(X)$ and $X$ has a non-constant meromorphic first integral by proposition 2. In particular, all leaves of $\mathcal{F}_{X}$ are closed and since $X$ is non-dicritical it has a non-constant holomorphic first integral by [14]. ∎

Remark 3.7**.**

When the origin is not an isolated singularity of $X\in\mathcal{X}_{2}$ then $d(X)$ can be arbitrarily large and finite. An example is $X_{k}=(x\,y)^{k}.\,R$, where $R=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ is the radial vector field. It can be checked that

[TABLE]

which has dimension $d(X)=2k+2$ (see also example 4 in § 2.1).**

This motivates the following:

Problem 5**.**

Let $X\in\mathcal{X}_{2}$ with $d(X)=\infty$ and non-isolated singularity at $0\in\mathbb{C}^{2}$. Is $\mathcal{I}(X)\neq\mathbb{C}$ ?**

Remark 3.8**.**

Example 10 with $n\geq 2$ has two distinguished separatrices: $(x=0)$ and $(y=0)$. In fact, consider the pencil of commuting vector fields $X_{\lambda}=x^{n}\frac{\partial}{\partial x}+\lambda\,y^{n}\frac{\partial}{\partial y}$, $\lambda\in\mathbb{C}^{*}$. Then $(x^{n-1}-\lambda\,y^{n-1})/x^{n-1}\,y^{n-1}$ is a first integral of $X_{\lambda}$. In particular, if $\lambda\neq 0,1$ then the unique commom separatrices of $X_{1}=X$ and $X_{\lambda}$ are the curves $(x=0)$ and $(y=0)$.

If $\lambda\neq 0$, when we blow-up once at $0\in\mathbb{C}^{2}$, $\Pi\colon(\widetilde{\mathbb{C}}^{2},D)\to(\mathbb{C}^{2},0)$, the strict transform $\widetilde{\mathcal{F}}_{\lambda}$ of the foliation defined by $\Pi^{*}(X_{\lambda})$ has n+1 singularities, all non-degenerated, two of them are non-dicritical and the others dicritical. The non-dicritical singularities correspond to the directions $(x=0)$ and $(y=0)$ and don’t change with the parameter. The dicritical singularities move along the divisor $D$ with the parameter $\lambda$. If $n=2$ the dicritical singularity is purely radial, whereas if $n>2$ then at any dicritical singularity the foliation $\widetilde{\mathcal{F}}_{\lambda}$ has a meromorphic first integral which in some cordinate system $(u,v)$ is of the form $v^{n-1}/u$, where the local equation of $D$ is $v=0$. This motivates the following:

Problem 6**.**

Is the above situation general? More specifically, suppose that $X\in\mathcal{X}_{2}$ is dicritical, has an isolated singularity at $0\in\mathbb{C}^{2}$, $\mu(X,0)\geq 2$ and $r(X)=2$. Let $Y\in C(X)$ such that $X\wedge Y\neq 0$ and consider the pencil $\lambda\mapsto X_{\lambda}:=X+\lambda\,Y$.

$1^{st}$ question: is there a blowing-up process $\Pi\colon(M,E)\to(\mathbb{C}^{2},0)$, where $E$ has an irreducible component $D$ which contains movables non-degenerated singularities with local meromorphic first integrals?

$2^{nd}$ question: does there exists a holomorphic family $\lambda\mapsto f_{\lambda}$ of non-constant meromorphic first integrals: $X_{\lambda}(f_{\lambda})=0$?**