Local normal forms of dynamical systems with a singular underlying geometric structure

TL;DR

This paper establishes conditions under which local normal forms can be simultaneously achieved for vector fields and singular geometric structures, extending classical normalization results to broader singular contexts.

Contribution

It introduces a unified approach for local normalization of vector fields coupled with various singular geometric structures, including volume, symplectic, Poisson, and contact forms, using toric methods.

Findings

Existence of formal simultaneous normalization for singular structures and vector fields.

Analytic normalization when structures and vector fields are analytic and integrable.

Application of toric and equivariant methods to normalization problems.

Abstract

In this paper we prove the existence of a simultaneous local normalization for couples $(X,\mathcal{G})$, where $X$ is a vector field which vanishes at a point and $\mathcal{G}$ is a singular underlying geometric structure which is invariant with respect to $X$, in many different cases: singular volume forms, singular symplectic and Poisson structures, and singular contact structures. Similarly to Birkhoff normalization for Hamiltonian vector fields, our normalization is also only formal, in general. However, when $\mathcal{G}$ and $X$ are (real or complex) analytic and $X$ is analytically integrable or Darboux-integrable then our simultaneous normalization is also analytic. Our proofs are based on the toric approach to normalization of dynamical systems, the toric conservation law, and the equivariant path method. We also consider the case when $\mathcal{G}$ is singular but $X$ does not vanish at the origin.

Equations

$$X=X^s+X^n+\sum _{i\ge 2}{X}^{(i)},$$

$$X^s=\sum _{i=1}^n{\gamma }_i{x}_i\frac{\partial }{\partial x_i},$$

$$[X^s,X]=0,$$

$$X=X^S+X^N$$

$$X^S=\sum _{i=1}^{\tau }{\rho }_i{Z}_i,$$

$$Z_i=\sum _{j=1}^n{a}_{ij}{x}_j\frac{\partial }{\partial x_j}$$

$$F=\prod _{k=1}^m{F}_k^{c_k},$$

$$\frac{dF}{F}=d(\ln F)=\sum _{k=1}^m{c}_k\frac{d{F}_k}{F_k}$$

$${\mathcal{L}}_X\Omega =\lambda \Omega \text{and}X(f)=\lambda f$$

$${\mathcal{L}}_{X^{(1)}}{\Omega }^{(\ell )}+\sum _{k=2}^{\ell +1}{\mathcal{L}}_{X^{(k)}}{\Omega }^{(\ell +1-k)}={\lambda }^{(0)}{\Omega }^{(\ell )}+\sum _{k=1}^{\ell }{\lambda }^{(k)}{\Omega }^{(\ell -k)}.$$

$${\mathcal{L}}_{X^{(1)}}{\mathcal{L}}_{X^s}{\Omega }^{(\ell )}+{\lambda }^{(0)}\sum _{k=2}^{\ell +1}{\mathcal{L}}_{X^{(k)}}{\Omega }^{(\ell +1-k)}={\lambda }^{(0)}{\mathcal{L}}_{X^s}{\Omega }^{(\ell )}+\lambda (0)\sum _{k=1}^{\ell }{\lambda }^{(k)}{\Omega }^{(\ell -k)}.$$

$${\mathcal{L}}_{X^{(1)}}\left({\mathcal{L}}_{X^s}{\Omega }^{(\ell )}-{\lambda }^{(0)}{\Omega }^{(\ell )}\right)={\lambda }^{(0)}({\mathcal{L}}_{X^s}{\Omega }^{(\ell )}-{\lambda }^{(0)}{\Omega }^{(\ell )});$$

$${\mathcal{L}}_{X^s}\left({\mathcal{L}}_{X^s}{\Omega }^{(\ell )}-{\lambda }^{(0)}{\Omega }^{(\ell )}\right)={\lambda }^{(0)}({\mathcal{L}}_{X^s}{\Omega }^{(\ell )}-{\lambda }^{(0)}{\Omega }^{(\ell )}).$$

$${\mathcal{L}}_{X^s}({\Omega }^{(r+j)})+{\mathcal{L}}_{X^n}({\Omega }^{(r+j)})+{\mathcal{L}}_{X^{(2)}}({\Omega }^{(r+j-1)})+\dots +{\mathcal{L}}_{X^{(j+1)}}({\Omega }^{(r)})={\lambda }^{(0)}{\Omega }^{(r+j)}+\dots +{\lambda }^{(j)}{\Omega }^{(r)}$$

$$\tilde{\Omega }={\Omega }^{(r)}+\cdots +{\Omega }^{(r+k-1)}+({\Omega }^{(r+k)}+{\Omega }^{(r)}{\beta }_k)+\cdots$$

$${\mathcal{L}}_X(\tilde{\Omega })=\tilde{\lambda }\tilde{\Omega },$$

$$\tilde{\lambda }={\lambda }^{(0)}+\cdots +{\lambda }^{(k-1)}+({\lambda }^{(k)}+X^{(1)}({\beta }_k))+\cdots .$$

$${\phi }_t^{\ast }\mathcal{G}(t)={\mathcal{G}}_0,\forall t\in [0,1].$$

$${\mathcal{L}}_{X_t}\mathcal{G}(t)+\frac{d}{dt}\mathcal{G}(t)=0$$

$$\frac{d}{dt}{\phi }_t^{\ast }\mathcal{G}(t)={\phi }_t^{\ast }\left({\mathcal{L}}_{X_t}\mathcal{G}(t)+\frac{d}{dt}\mathcal{G}(t)\right)=0,$$

$$Y_t={\int }_G{\rho }_g^{\ast }{X}_{t}d\mu ,$$

$${\mathcal{L}}_{Y_t}\mathcal{G}(t)+\frac{d}{dt}\mathcal{G}(t)$$

$$={\int }_G\left({\mathcal{L}}_{{\rho }_g^{\ast }{X}_t}{\rho }_g^{\ast }\mathcal{G}(t)+\frac{d}{dt}{\rho }_g^{\ast }\mathcal{G}(t)\right)d\mu$$

$$={\int }_G{\rho }_g^{\ast }\left({\mathcal{L}}_{X_t}\mathcal{G}(t)+\frac{d}{dt}\mathcal{G}(t)\right)d\mu =0,$$

$$\Omega =f(x_1,\dots ,x_n)d{x}_1\wedge \dots \wedge d{x}_n$$

$$\Lambda =g(x_1,\dots ,x_n)\frac{\partial }{\partial x_1}\wedge \dots \wedge \frac{\partial }{\partial x_n},$$

$$\Omega =fd{x}_1\wedge \dots \wedge d{x}_n$$

$$\Omega =y_{1}d{y}_1\wedge \dots \wedge d{y}_n$$

$$\Omega =fd{\hat{x}}_1\wedge d{x}_2\wedge \dots \wedge d{x}_n$$

$$g(x_1,\dots ,x_n)={\int }_0^{x_1}t\varphi (t,x_2,\dots ,x_n)dt$$

$$\Omega =dg\wedge d{x}_2\wedge \dots \wedge d{x}_n,$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Associated torus actions

4. 2.2 Integrable systems

5. 2.3 The toric conservation law

6. 2.4 Equivariant Moser path method

7. 3 Systems with an invariant singular volume form

8. 3.1 The case of a folded volume form

9. 3.2 Systems preserving a singular Nambu structure of top order

10. 4 Systems with a singular symplectic structure

11. 4.1 Folded symplectic structures

12. 4.2 Log-symplectic structures

13. 4.3 Equivariant normalization of log-symplectic structures

14. 5 Systems with a singular contact structure

15. 5.1 Non-vanishing singular contact forms

16. 5.1.1 The case $X(O)\neq 0$

17. 5.1.2 The case $X(O)=0$

18. 5.2 Two types of non-degenerate singular contact forms

19. 5.2.1 The transversal case

20. 5.2.2 The tangential case

21. 5.2.3 Singular contact forms with integrable systems

1. Introduction

A classical result due to Birkhoff, Gustavson, and Moser (see, e.g., [4, 16, 27]) states that a Hamiltonian vector field $X$ which vanishes at a point $O$ admits a normalization à la Poincaré-Birkhoff at $O$ (also in the resonant case), i.e., there is a coordinate system $(x_{1},\ldots,x_{2n})$ in which $X$ does not contain any non-resonant terms and the symplectic form has the canonical expression $\omega=\sum_{i=1}^{n}dx_{i}\wedge dx_{i+n}$. A similar result, with a similar proof, is known to hold for volume-preserving vector fields: if $X$ preserves a volume form $\Omega$ then there is a coordinate system $(x_{1},\ldots,x_{n})$ which normalizes $X$ and in which $\Omega$ has the canonical expression $\Omega=dx_{1}\wedge\ldots\wedge dx_{n}$. These normalizations of Hamiltonian and volume-preserving vector fields are only formal, in general. However, when $X$ is integrable analytically or in the sense of Darboux, then there exists a locally analytic normalization [32, 37, 38, 41].

The problem of normalization of Hamiltonian and volume-preserving vector fields may be viewed as the problem of simultaneous normalization of a couple $(X,\mathcal{G})$, where $\mathcal{G}$ is an underlying geometric structure preserved by $X$. (In the above cases, $\mathcal{G}$ is a symplectic or a volume form.) In this paper, we study this simultaneous normalization problem, when the geometric structure $\mathcal{G}$ itself is singular at the singular point $O$ of $X$. We may encounter singularities of different types and some singular structures have been studied by different people from various aspects, see, e.g., [20, 29, 28, 35, 18, 25]. We concentrate our attention to the following list of singular geometric structures, though many other structures (e.g., more general Poisson, Dirac, or Nambu structures) appear in geometric mechanics and physics:

$\bullet$ Singular volume forms, which either vanish or blow up at the singular point $O$ of $X$ (e.g., $x_{1}^{k}dx_{1}\wedge\ldots\wedge dx_{n}$ where $k=\pm 1$).

$\bullet$ Singular symplectic forms, which are either folded symplectic or log-symplectic (e.g., $\sum x_{i}^{k_{i}}dx_{i}\wedge dx_{n+i}$ where $k_{i}\in\{-1,0,1\}$).

$\bullet$ Singular contact structures.

In each of the above cases, we impose some non-degeneracy or genericity condition ensuring that the singular geometric structure itself admits a nice normal norm, i.e., has some local canonical expression, e.g., $\dfrac{1}{x_{1}}dx_{1}\wedge dx_{n+1}+\sum_{i=2}^{n}dx_{i}\wedge dx_{n+i}$ for a so-called $b$-symplectic structure [12, 13, 14], or analogous expressions for $b^{m}$-symplectic and $b^{m}$-Nambu structures [22, 23] (even in the equivariant case, if needed). Then we show that, in each of these cases, the couple $(X,\mathcal{G})$ admits a simultaneous normalization, i.e., a coordinate system which puts $\mathcal{G}$ in some canonical form and $X$ in normal form.

The simultaneous normalization problem is usually quite difficult, especially when the underlying geometric structure itself is singular. As far as we know, there are two main methods to study it.

The first approach is classical: one first puts the geometric structure $\mathcal{G}$ in canonical form, then makes a step-by-step normalization of the vector field $X$ (eliminating resonant nonlinear terms one by one) by “canonical transformations” which leave $\mathcal{G}$ intact. This approach was effective for Hamiltonian and volume-preserving systems and can probably be used also for other systems. However, to show the existence of the required “canonical transformations” at each step is a non-trivial task, especially when the geometric structure is itself singular. For example, when the singular structure is not homogeneous, one cannot find canonical transformations order by order directly. In order to leave the geometric structure unchanged, one needs to adjust, at each step, the terms of the transformation having different orders; this results in very long, laborious, and delicate computations.

The second approach, initiated by the third author [37, 38, 25, 40], is geometric and is based on the toric characterization of the normalization of dynamical systems, together with the equivariant Moser path method [26] (a.k.a. the Lie transform method). This is the approach used in this paper and it goes as follows.

$\bullet$ For each vector field (or family of commuting vector fields) which vanishes at a point $O$ there is a unique natural (intrinsic) local effective associated torus $\mathbb{T}^{\tau}$-action $\rho$ (in the complexified space, if the original space is real) which fixes $O$. The dimension $\tau$ of the torus in question is called the toric degree of the system at $O$. This action $\rho$ is only formal, in general, but when the system is analytically or Darboux integrable, then it is automatically analytic.

$\bullet$ Universal toric conservation law: any geometric structure $\mathcal{G}$ preserved by a dynamical system is also preserved by its associated torus action. This is one of our main results and it applies to many other geometric structures, not just the ones studied in this paper; see Theorem 1.1 below for a precise statement.

$\bullet$ For the geometric structures studied in this paper, it is known, or it can be shown, that $\mathcal{G}$ can be put into canonical form in some coordinate system (if one forgets about $\rho$), and it is also known that $\rho$ can be linearized by Bochner’s averaging formula [5] (if one forgets about $\mathcal{G}$). What needs to be done do is to linearize $\rho$ and put $\mathcal{G}$ in canonical form simultaneously which is achieved by the use of the equivariant Moser path method.

$\bullet$ The linearization of $\rho$ is actually equivalent to the normalization of the dynamical system. So, the previous step produces a coordinate system in which both the geometric structure and the dynamical system are normalized.

In general, the above steps only give a formal simultaneous normalization, just like in the classical theory of Poincaré-Birkhoff normalization of vector fields. The problem of showing the existence, or non-existence, of a local analytic normalization is, in general, difficult since it involves small divisor phenomena which are hard to control; see, e.g., [7, 17, 31]. However, when our system is analytically or Darboux integrable, then the associated torus action is analytic, the path method can also be done analytically, and we obtain an analytic simultaneous normalization without having to deal with small divisors.

Our approach (as well as the first classical one) works mainly in the formal and analytic categories, so our results address predominantly formal and analytic systems, though some results are also valid in the smooth category. (The proofs in the smooth category are usually more involved and require some specific techniques.)

In this paper, most of the times we do not explicitly distinguish between the real and complex cases. Our results are valid for both complex and real systems, i.e., in the real case the normalization maps are also real, though we only write explicit proofs for the complex case. The reason is that, by local complexification, a real dynamical system (together with an underlying geometric structure) can be viewed as a complex dynamical system which is equivariant with respect to the anti-complex involution. As was explained in, e.g., [15] and [38], even if we deal with a real system, everything can be done in the complexified space by keeping track of the anti-complex involution, so that the complex normalization maps in the real case turn out to be equivariant with respect to the anti-complex involution, i.e., they are, in fact, complexifications of real normalization maps. Of course, some “finer” normal forms, e.g., the Jordan normal form for matrices, look differently in complex and real coordinates, but this is not the main point of our paper.

The paper is organized as follows. In Section 2, we collect necessary notions and tools to be used for our study, including the toric approach to the normalization problem of vector fields (Subsection 2.1), the notions of analytic and Darboux integrability (Subsection 2.2), the universal toric conservation law (Subsection 2.3), and the equivariant Moser path method (Subsection 2.4). The main new result of Section 2 is the following theorem, an important particular case (which has not been proved explicitly anywhere, as far as we know) of the universal toric conservation law stating that “anything which is preserved by a dynamical system is also preserved by its associated torus actions” [40].

Theorem 1.1** (Theorem 2.3).**

Let $X$ be a formal vector field which vanishes at a point $O$ and which preserves a formal rational tensor field $\Lambda$ (i.e., $\Lambda=\Omega/f$ where $\Omega$ is a formal tensor field and $f$ is a non-trivial formal function). Then the associated torus action of $X$ at $O$ also preserves $\Lambda$. If $\Lambda$ is only conformally preserved by $X$ (i.e., $\mathcal{L}_{X}\Lambda=g\Lambda$ where $g$ is a formal function), then the associated torus action of $X$ also conformally preserves $\Lambda$.

The case with an invariant singular volume form is treated in Section 3, with an invariant singular symplectic form in Section 4, and with an invariant singular contact structure in Section 5. The main results of these last three sections can be put together in the following big abstract theorem.

Theorem 1.2**.**

Let $X$ be a formal vector field which vanishes at the origin and which preserves a singular geometric structure $\mathcal{G}$ which belongs to one of the following types: folded volume forms, generic singular Nambu structures of top order, (multi-)folded symplectic forms, log-symplectic forms, and three different kinds of singular contact structures. Then the couple $(X,\mathcal{G})$ can be formally normalized simultaneously. Moreover, in the integrable analytic case, when both $\mathcal{G}$ and $X$ are analytic and $X$ is analytically or Darboux integrable, then there exists a local analytic simultaneous normalization of $(X,\mathcal{G})$.

We also treat the cases of couples $(X,\mathcal{G})$ where the geometric structure is singular at the origin $O$ but $X(O)\neq 0$. In a series of theorems throughout this paper, we show that, in such situations, $\mathcal{G}$ can often be put into normal form in a coordinate system in which the vector field $X$ becomes $\partial/\partial x_{k}$ where $x_{k}$ is one of the coordinates. These theorems (for the cases $X(O)\neq 0$) use more classical step-by-step normalization methods and not the toric approach for $X$.

2. Preliminaries

2.1. Associated torus actions

In this subsection, we briefly recall from [37, 38, 39, 40] the toric approach to the problem of local normalization of vector fields.

Denote by $X$ a formal or local analytic vector field on a manifold, which vanishes at a point $O$. We write the Taylor series of $X$ in a coordinate system $(x_{1},\ldots,x_{n})$ around $O$ as

[TABLE]

where $X^{(1)}=X^{s}+X^{n}$ is the Jordan decomposition of the linear part of $X$ in the above coordinate system, i.e. $X^{s}$ is its semisimple part and $X^{n}$ is its nilpotent part, and $X^{(i)}$ is the homogeneous term of degree $i$ of $X$. We can assume (after a complexification of the system, if necessary) that $X^{s}$ is diagonal:

[TABLE]

where $\gamma_{1},\ldots,\gamma_{n}\in\mathbb{C}$ are the eigenvalues of $X$ at $O$. We say that $X$ is in normal form (à la Poincaré–Birkhoff or Poincaré–Dulac) if

[TABLE]

which amounts to the vanishing of all the non-resonant terms in the Taylor expansion of $X$. The classical theorem of Poincaré says that such a formal coordinate system always exists, i.e., $X$ can always be normalized formally.

We may view $X$, via its Lie derivative, as a linear differential operator on the space of formal functions. Then this operator, even though acting on an infinite dimensional space, also admits an intrinsic unique Jordan decomposition

[TABLE]

where $X^{S}$ and $X^{N}$ are formal vector fields (also viewed as linear operators) which are intrinsic (i.e., they depend on $X$ but do not depend on the choice of local coordinates). The vector field $X$ is in normal form if and only if $X^{S}$ is linear semisimple, i.e. $X^{S}=X^{s}$ and $X^{N}=X^{n}+X^{(2)}+\cdots$.

The smallest number $\tau\geq 0$ such that we can express $X^{S}=X^{s}$ (in normalized coordinates) in the form

[TABLE]

where $\rho_{i}\in\mathbb{C}$ are constants and

[TABLE]

are diagonal vector fields with integer coefficients $a_{ij}\in\mathbb{Z}$, is called the toric degree of $X$ at $O$. The minimality condition on $\tau$ is equivalent to the condition that $\rho_{1},\ldots,\rho_{\tau}$ are incommensurable. The vector fields $\sqrt{-1}Z_{1},\ldots,\sqrt{-1}Z_{\tau}$ are the generators of an effective torus $\mathbb{T}^{\tau}$-action $\rho$, which is uniquely determined by $X$ (up to automorphisms), and which is called the intrinsic associated torus action of $X$ at $O$.

In general, even when $X$ is analytic, its normalization is only formal, and its associated torus action $\rho$ is also formal. Thanks to Bochner’s averaging formula [5], any compact group action (even formal ones) can be linearized, and the normalization of $X$ is the same as the linearization of its associated torus action $\rho$ (i.e., $X$* is in normal form if and only if $\rho$ is linear*). The vector field $X$ admits a local analytic normalization if and only if the action $\rho$ is analytic. So this torus action $\rho$ plays a very important role in normalization problems and it admits the fundamental conservation property [40] which will be discussed in Subsection 2.3.

2.2. Integrable systems

Recall that a Darboux-type function is a (generally multi-valued) function $F$ which can be written as

[TABLE]

where $F_{k}$ are analytic functions and $c_{k}$ are complex numbers. The class of Darboux-type functions on a manifold is significantly larger than the class of analytic or rational functions. Note that, even though $F=\prod_{k=1}^{m}F_{k}^{c_{k}}$ is multivalued, its logarithmic differential

[TABLE]

is a single-valued rational differential 1-form, so it makes sense to talk about Darboux-type first integrals of vector fields: $X(\prod_{k=1}^{m}F_{k}^{c_{k}})=0$ means that $\sum_{k=1}^{m}c_{k}\dfrac{X(F_{k})}{F_{k}}=0$. In many problems in dynamical systems, one may be interested in Darboux-type first integrals when there do not exist enough analytic or rational first integrals (see., e.g., [36]).

In this paper, for analytic normalization, we will consider dynamical systems which are integrable in the usual non-Hamiltonian sense (see, e.g., [6, 31, 37, 40]), also in the case when the first integrals are only Darboux-type functions. Let us recall here the definition of integrability.

Definition 2.1**.**

i) A dynamical system given by a vector field $X$ on an $n$-dimensional manifold $M$ is called integrable if there exist $p$ vector fields $X_{1}=X$, $X_{2},\ldots,X_{p}$ and $q$ functions $F_{1},\ldots,F_{q}$ on $M$, such that the vector fields commute pairwise, $X_{1}\wedge\cdots\wedge X_{p}\neq 0$ almost everywhere, the functions are common first integrals for these vector fields, and $\mathrm{d}F_{1}\wedge\cdots\wedge\mathrm{d}F_{q}\neq 0$ almost everywhere. The integers $p,q$ satisfy $p\geqslant 1,q\geqslant 0,p+q=n$.

ii) An $n$-tuple $(X_{1},\ldots,X_{p},F_{1},\ldots,F_{q})$ such as above is also called an integrable system of type $(p,q)$.

iii) If the vector fields $X_{1},\ldots,X_{p}$ and the functions $F_{1},\ldots,F_{q}$ are all smooth (resp. formal, resp. analytic) then the vector field $X$ and the system $(X_{1},\ldots,X_{p},F_{1},\ldots,F_{q})$ are called smoothly (resp. formally, resp. analytically) integrable.

iv) If the vector fields $X_{1},\ldots,X_{p}$ are rational (i.e. can be written as the quotient of an analytic vector field by a non-trivial analytic function), and the functions $F_{1},\ldots,F_{q}$ are Darboux-type functions, then $X$ and the system $(X_{1},\ldots,X_{p},F_{1},\ldots,F_{q})$ are called Darboux integrable.

The following result, obtained by the third author via geometric approximation methods (instead of the fast convergence method), will allow us to analytically normalize integrable systems.

Theorem 2.2** ([37, 38, 41]).**

Let $X$ be a local analytic vector field which vanishes at a point $O$ and which is analytically integrable or Darboux integrable. Then the associated torus action of $X$ at $O$ is locally analytic.

2.3. The toric conservation law

The toric conservation law, which states that anything which is preserved by a dynamical system is also preserved by its associated torus action, was first discovered in its general form by the third author [40], though some particular cases of this law (e.g., when that “anything” is a function, i.e., a first integral) have been known before. Here, the system may be given by a vector field, or a discrete-time system, or a quantum system, or a stochastic system, etc.; “anything” may be a tensor field, or a subbundle of a natural bundle over the manifold, or a differential operator, etc., which may be smooth, analytic, meromorphic, formal, Darboux-type, etc.. The associated torus actions in question depend on the situation: for example, for integrable Hamiltonian systems near a regular Liouville torus, this torus action is the Liouville torus action, whereas if the system is a local dynamical system which vanishes at a point, then this torus action is the one explained in Subsection 2.1.

For each particular situation, the toric conservation law can be stated as a rigorous theorem or conjecture. There are still many situations for which no proof of the corresponding conjectures have been explicitly written. In this section we consider such a situation, which is useful for our simultaneous normalization problems.

Consider a formal vector field $X$ on $\mathbb{C}^{n}$ which vanishes at the origin and let $\Lambda$ be a formal rational tensor field on $\mathbb{C}^{n}$, i.e., $\Lambda$ is the quotient of a formal tensor field $\Omega$ by a nontrivial formal function $f$: $\Lambda=\dfrac{\Omega}{f}$. We say that $\Lambda$ is an invariant of $X$ if it is conserved by $X$, i.e., $\mathcal{L}_{X}\Lambda=0$. We say that $\Lambda$ is a semi-invariant of $X$ if it is conformally conserved by $X$, i.e., there is a formal function $g$ such that $\mathcal{L}_{X}\Lambda=g\Lambda$.

Observe that the set of nontrivial semi-invariant formal rational functions of $X$ form a multiplicative group: if $\mathcal{L}_{X}F_{1}=g_{1}F_{1}$ and $\mathcal{L}_{X}F_{2}=g_{2}F_{2}$ then $\mathcal{L}_{X}(1/F_{1})=-g_{1}(1/F_{1})$ and $\mathcal{L}_{X}(F_{1}F_{2})=(g_{1}+g_{2})F_{1}F_{2}$. Moreover, if $\Lambda=\dfrac{\Omega}{f}$ is a formal rational tensor field written in reduced form, i.e. $\Omega$ and $f$ are co-prime, and $\Lambda$ is a semi-invariant tensor field of $X$, then both $\Omega$ and $f$ are semi-invariants of $X$. Indeed, $\mathcal{L}_{X}(\Omega/f)=g\Omega/f$ means that $f\mathcal{L}_{X}(\Omega)-\mathcal{L}_{X}(f)\Omega=gf\Omega$, or $\mathcal{L}_{X}(f)\Omega=f(\mathcal{L}_{X}\Omega-g\Omega)$. Since $\Omega$ is co-prime with $f$, it implies that $\mathcal{L}_{X}(f)$ is divisible by $f$, i.e., $\mathcal{L}_{X}(f)=hf$, where $h$ is a formal function, and then we have $\mathcal{L}_{X}\Omega=(g+h)\Omega$.

Theorem 2.3** (Conservation Theorem).**

Let $X$ be a formal vector field on $\mathbb{C}^{n}$ which vanishes at the origin. Assume that $X$ is already in Poincaré-Dulac normal form and that $(Z_{1},\ldots,Z_{\tau})$ is a $\tau$-tuple of diagonal vector fields which generate the associated torus action of $X$ as in (2.1), (2.2). Let $\Lambda$ be a formal rational tensor field on $\mathbb{C}^{n}$. Then we have:

(i)* If $\Lambda$ is preserved by $X$, i.e., $\mathcal{L}_{X}\Lambda=0$, then $\Lambda$ is also preserved by the associated torus action of $X$, i.e., $\mathcal{L}_{Z_{k}}\Lambda=0$ for every $k=1,\ldots,\tau$.*

(ii)* If $\Lambda$ is conformally preserved by $X$, i.e., $\mathcal{L}_{X}\Lambda=g\Lambda$, where $g$ is a formal function, then $\Lambda$ is also conformally preserved by the associated torus action of $X$, i.e., $\mathcal{L}_{Z_{k}}\Lambda=g_{k}\Lambda$ for every $k=1,\ldots,\tau$, where $g_{1},\ldots,g_{\tau}$ are formal functions.*

In order to prove the above theorem, we recall the following result due to Walcher (Lemma 2.2 in [33], see also Lemma 2.1 of [41]).

Lemma 2.4**.**

Let $X$ be a formal vector field in Poincaré-Dulac normal form. Assume that $F=\sum_{k}F^{(k)}$ is a formal semi-invariant invariant of $X$, i.e., $X(F)=\lambda F$ for some formal function $\lambda$. Then there exists an invertible formal function $\beta$ (i.e., $\beta(0)\neq 0$) such that $\tilde{F}=\beta F$ is a semi-invariant of both $X$ and of its semisimple part $X^{s}$. Moreover, the formal function $\tilde{\lambda}$, satisfying $X(\tilde{F})=\tilde{\lambda}\tilde{F}$, is a first integral of $X^{s}$ and we have $X^{s}(\tilde{F})=\lambda(0)\tilde{F}$. In addition, $Z_{k}(\tilde{F})=c_{k}\tilde{F}$ for every $k=1,\ldots,\tau$, where $c_{k}$ are some complex numbers and $Z_{1},\ldots,Z_{\tau}$ are the generators given in formulas (2.1) and (2.2) of the associated torus action of $X$.

We now turn to the proof of the Conservation Theorem 2.3.

Proof.

(i) We can write the formal rational tensor field $\Lambda$ as $\Lambda=\Omega/f$ where $f$ is a formal function and $\Omega$ is a formal tensor field co-prime with $f$. As observed above, $\mathcal{L}_{X}(\Omega/f)=0$ implies that $f$ and $\Omega$ are semi-invariants of $X$, i.e.,

[TABLE]

for some formal function $\lambda$. Invoking Walcher’s Lemma 2.4, we can assume $\lambda$ is a first integral of $X^{s}$ (by multiplying both $\Omega$ and $f$ by a formal function of the type $1+\beta$, if necessary), that $X^{s}(f)=\lambda(0)f$, and that $Z_{k}(f)=c_{k}f$ with $c_{k}\in\mathbb{C}$ for $k=1,\ldots,\tau$.

We want to show that $\mathcal{L}_{X^{s}}(\Omega/f)=0$ or, equivalently, $f\mathcal{L}_{X^{s}}\Omega=X^{s}(f)\Omega$. Since $X^{s}(f)=\lambda(0)f$, the equation to prove is reduced to $\mathcal{L}_{X^{s}}\Omega=\lambda(0)\Omega$. We prove it by induction on the degree of the terms in (2.3).

Write $\lambda=\lambda^{(0)}+\lambda^{(1)}+\cdots$ and $\Omega=\Omega^{(0)}+\Omega^{(1)}+\cdots$ in which the upper right index $(k)$ indicates the homogeneous part of degree $k$ of the corresponding term. Obviously, $\mathcal{L}_{X^{(1)}}\Omega^{(0)}=\lambda(0)\Omega^{(0)}$, therefore $\mathcal{L}_{X^{s}}\Omega^{(0)}=\lambda(0)\Omega^{(0)}$. Now suppose $\mathcal{L}_{X^{s}}\Omega^{(k)}=\lambda^{(0)}\Omega^{(k)}$ for all $k<\ell$. Look at the terms of degree $\ell$ in (2.3); we have

[TABLE]

Let $X^{s}$ act on both sides of the above equation. Since $[X^{s},X^{(k)}]=0$ and $X^{s}(\lambda^{(k)})=0$, we have $\mathcal{L}_{X^{s}}\Omega^{(k)}=\lambda(0)\Omega^{(k)}$ for all $k$ by Lemma 2.4, and so we get

[TABLE]

Take the difference between the expression (2.5) and $\lambda^{(0)}$ times the expression (2.4) to get

[TABLE]

therefore

[TABLE]

This means that $\mathcal{L}_{X^{s}}\Omega^{(\ell)}-\lambda^{(0)}\Omega^{(\ell)}$ is an eigenvector with eigenvalue $\lambda^{(0)}$ of the linear operator $\mathcal{L}_{X^{s}}$ on the space $E$ of homogeneous tensor fields of degree $\ell$. We now decompose $E$ into a direct sum of eigenspaces and write $\Omega^{(\ell)}=\sum_{j}\Omega^{(\ell)}_{j}$ where $\Omega^{(\ell)}_{j}$ is a vector in the eigenspace $E_{c_{j}}$ with eigenvalue $c_{j}$. Restricting attention to $E_{c_{j}}$, we can easily conclude $c_{j}=\lambda^{(0)}$ by (2.6). Thus $c_{j}=\lambda^{(0)}$ for all $j$, and $\Omega^{(\ell)}$ itself lies in the eigenspace $E_{\lambda^{(0)}}$, that is, $\mathcal{L}_{X^{s}}\Omega^{(\ell)}=\lambda^{(0)}\Omega^{(\ell)}$. By induction, we conclude that $\mathcal{L}_{X^{s}}\Omega=\lambda^{(0)}\Omega$.

Each monomial tensor field $\Theta$ is an eigenvector of the Lie derivative operator of the diagonal vector field $X^{s}=\sum_{k=1}^{\tau}\rho_{k}Z_{k}$, i.e., we have $\mathcal{L}_{X^{s}}\Theta=\sum_{k=1}^{\tau}\rho_{k}c_{k}(\Theta)\Theta$ for some numbers $c_{k}(\Theta)$ which, a priori, depend on $\Theta$. Since $\mathcal{L}_{X^{s}}\Omega=\lambda^{(0)}\Omega$, we must have $\sum_{k=1}^{\tau}\rho_{k}c_{k}(\Theta)=\lambda(0)$ for every monomial term $\Theta$ of $\Omega$. Since the numbers $\rho_{1},\ldots,\rho_{\tau}$ are incommensurable, there is at most one $\tau$-tuple of complex numbers $c_{1},\ldots,c_{k}$ satisfying the equation $\sum\rho_{k}c_{k}=\lambda(0)$, which means that $c_{k}=c_{k}(\Theta)$ does not depend on $\Theta$ and that $\mathcal{L}_{Z_{k}}(\Omega)=c_{k}\Omega$. Similarly, we have $\mathcal{L}_{Z_{k}}(f)=c_{k}f$ and hence $\mathcal{L}_{Z_{k}}(\Omega/f)=0$.

(ii) The proof of the second part consists of the following 3 steps:

Step 1. Write $\Lambda=\Omega/f$, where $\Omega$ and $f$ are co-prime. Then both $\Omega$ and $f$ are semi-invariants of $X$. By Lemma 2.4, we conclude that $f$ is a semi-invariant of the associated torus action of $X$, so it is enough to show that $\Omega$ is also a semi-invariant of the associated torus action of $X$, i.e., the problem is reduced to the case when $\Lambda=\Omega$ is a formal tensor field.

Step 2. Write $\mathcal{L}_{X}\Omega=\lambda\Omega$, where $\lambda$ is a formal function. Putting $\Omega^{*}=\beta\Omega$, where $\beta$ is an appropriate invertible formal function, we can arrange so that $\mathcal{L}_{X^{s}}\Omega^{*}=\lambda(0)\Omega^{*}$. The proof of this step is the same as the proof of Walcher’s Lemma 2.4. (See the proof of Lemma 2.1(i) in [41].) To keep the paper self-contained, due to the importance of the Conservation Theorem 2.3, we give below the full proof.

The semi-invariance of $\Omega$ with respect to $X$ is equivalent to

[TABLE]

for all $j\geq 0$.

Note that $X^{s}(\lambda^{(0)})=0$. Now assume that $X^{s}(\lambda^{(j)})=0$ for all $j<k$ and let $\tilde{\Omega}=(1+\beta_{k})\Omega$, where $\beta_{k}$ is some homogeneous function of degree $k$. Then

[TABLE]

and

[TABLE]

with

[TABLE]

Due to the semi-simplicity of $X^{s}$, one can choose $\beta_{k}$ such that $X^{s}(\lambda^{(k)}+X^{(1)}(\beta_{k}))=0$. Thus, the assertion is proved by induction on $k$: $\beta$ can be constructed in the form of an infinite product $\prod_{k=1}^{\infty}(1+\beta_{k})$, where each $\beta_{k}$ is homogeneous of degree $k$. (Such an infinite product converges in the space of formal power series). From $X^{s}(\lambda^{*})=0$ one deduces that $\mathcal{L}_{X^{s}}(\Omega^{*})=\lambda^{(0)}\Omega^{*}$, again by induction and by the semi-simplicity of $X^{s}$.

Step 3. The equation $\mathcal{L}_{X^{s}}(\Omega^{*})=\lambda^{(0)}\Omega^{*}$ implies that $\Omega^{*}$ (and hence $\Omega$) is a semi-invariant of the vector fields $Z_{k}$, $k=1,\ldots,\tau$. This step is already done at the end of the proof of (i). ∎

2.4. Equivariant Moser path method

We recall the well-known path method, which was introduced by Moser in [26] to show the equivalence of two volume forms, and which can be generalized to many other situations in order to show the equivalence of two tensor fields on a manifold (see, e.g., [1, Section 5.4] for a general presentation).

Let two tensor fields $\mathcal{G}_{0}$ and $\mathcal{G}_{1}$ be given on a smooth manifold $M$. We say they are locally equivalent at $O\in M$ if there is a diffeomorphism $\varphi$ of one neighborhood of $O$ to another neighborhood of $O$, such that $\varphi^{\ast}\mathcal{G}_{1}=\mathcal{G}_{0}$. One way to show that $\mathcal{G}_{0}$ and $\mathcal{G}_{1}$ are locally equivalent is to join them by a curve of tensor fields $\mathcal{G}(t)$ satisfying $\mathcal{G}(0)=\mathcal{G}_{0}$, $\mathcal{G}(1)=\mathcal{G}_{1}$ and to seek a curve of local diffeomorphisms $\varphi_{t}$ such that $\varphi_{0}=Id$ and

[TABLE]

Then $\varphi=\varphi_{1}$ is the desired diffeomorphism.

A way to find the curve of diffeomorphisms $\varphi_{t}$ satisfying the relation above is to solve the equation

[TABLE]

for a smooth time-dependent vector field $X_{t}$. If this is possible, let $\varphi_{t}=F_{t,0}$, where $F_{t,s}$ is the evolution operator of the time-dependent vector field $X_{t}$. Then we have

[TABLE]

so that $\varphi_{t}^{\ast}\mathcal{G}(t)=\varphi_{0}^{\ast}\mathcal{G}(0)=\mathcal{G}_{0}$. If we choose $X_{t}$ such that $X_{t}(O)=0$, then $\varphi_{t}$ exists for a time $t\geq 1$ in an open neighborhood of $O$ and $\varphi_{t}(O)=O$.

One often takes $\mathcal{G}(t)=(1-t)\mathcal{G}_{0}+t\mathcal{G}_{1}$. Also, in applications, this method is not always used in exactly this way since the algebraic equation for $X_{t}$ might be hard to solve. In such situations the spirit of the path method is used (as, e.g., in the proof of the Frobenius Theorem given in [1, Section 4.4]).

Now suppose that a compact Lie group $G$ acts smoothly on $M$ preserving both tensor fields $\mathcal{G}_{0}$ and $\mathcal{G}_{1}$, i.e., $\rho_{g}^{*}\mathcal{G}_{i}=\mathcal{G}_{i}$, for all $g\in G$, $i=0,1$, where $\rho:G\times M\rightarrow M$ is the $G$-action. Choose the path of tensor fields $\mathcal{G}(t)$ to be also invariant under the action $\rho$; this condition is clearly satisfied if $\mathcal{G}(t)=(1-t)\mathcal{G}_{0}+t\mathcal{G}_{1}$. Let $X_{t}$ be a solution of (2.7) and define the $G$-averaged smooth vector field

[TABLE]

where $\mu$ is the Haar measure on $G$. By invariance of the Haar measure relative to group translations, $Y_{t}$ is $G$-invariant. In addition, since $\rho_{g}^{*}\mathcal{G}(t)=\mathcal{G}(t)$ for any $g\in G$ and $t\in[0,1]$, we have

[TABLE]

and hence $Y_{t}$ also solves (2.7). This shows that $\mathcal{G}_{0}$ and $\mathcal{G}_{1}$ are $G$-equivariantly equivalent. We proved the following result.

Lemma 2.5**.**

Suppose that the compact Lie group $G$ acts locally smoothly (respectively formally or analytically) on $M$, preserving two tensor fields $\mathcal{G}_{0}$ and $\mathcal{G}_{1}$. If (2.7) has a smooth (respectively formal or analytic) solution $X_{t}$, then the two structures are locally smoothly (respectively formally or analytically) $G$-equivariantly equivalent.

3. Systems with an invariant singular volume form

In this section, we work with a local volume form

[TABLE]

which is singular in one of the following two senses: either $f$ is a formal or analytic function which vanishes at the origin $O=(0,\ldots,0)$, or $f$ blows up at $O$. When $f$ blows up at $O$, it is more convenient to look at the dual $n$-vector field (i.e., a contravariant volume form, a.k.a. Nambu structure of top order) $\Lambda=\Omega^{-1}$ defined by $\langle\Lambda,\Omega\rangle=1$, i.e,

[TABLE]

where $g=1/f$ is now a formal or analytic function which vanishes at $O$.

We assume that $O$ is a non-degenerate singular point, i.e., we have the following two cases:

• •

Case 1. $f(O)=0$ and $df(O)\neq 0$. In this case $\Omega$ is called a folded volume form (because it can be obtained as the pull-black of a usual volume form by a fold map, like in the case of folded symplectic structures [21]). This case is investigated in Subsection 3.1.

• •

Case 2. $g(O)=0$ and $dg(O)\neq 0$. In this case $\Omega$ is called a non-degenerate log-volume form and its dual $n$-vector field $\Lambda$ is a non-degenerate singular Nambu structure of top order. This case is investigated in Subsection 3.2.

3.1. The case of a folded volume form

The following lemma about the canonical form of a folded volume form is a well-known folklore result, but we don’t know any exact reference for it, so we present its proof.

Lemma 3.1**.**

Consider a local differential form of top order

[TABLE]

on an $n$-dimensional manifold $(n\geq 2),$ where $f$ is a formal (resp. local analytic, resp. smooth) function such that $f(O)=0$ and $df(O)\neq 0$. Then there exists a formal (resp. analytic, resp. smooth) coordinate system $(y_{1},\ldots,y_{n})$ around $O$ such that

[TABLE]

Proof.

Denote by $S=\{f=0\}$ the hypersurface of singular points of $\Omega$. By the implicit function theorem, locally we can write $S=\{x_{1}=h(x_{2},\ldots,x_{n})\}$ where $h$ is some formal (or analytic, or smooth) function. Put $\hat{x}_{1}=x_{1}-h(x_{2},\ldots,x_{n})$. Then

[TABLE]

and $f=0$ at $\hat{x}_{1}=0$. Thus, by this change of variables, we may assume that $S=\{x_{1}=0\}$, i.e, $f$ is divisible by $x_{1}$: $f=x_{1}\phi$, where $\phi$ is a function such that $\phi(O)\neq 0.$ Integrating $f=x_{1}\phi$ in the direction of $\dfrac{\partial}{\partial x_{1}}$, we obtain a function

[TABLE]

satisfying

[TABLE]

and such that $g(0,x_{2},\ldots,x_{n})=0$, $\dfrac{\partial g}{\partial x_{1}}(0,x_{2},\ldots,x_{n})=0$ but $\dfrac{\partial^{2}g}{\partial^{2}x_{1}}(O)=\phi(O)\neq 0$, which means that $g$ can be written as $g=x_{1}^{2}h$ where $h(0)\neq 0.$ Put $y_{1}=x_{1}\sqrt{2h}$ (so that $g=y_{1}^{2}/2$), $y_{2}=x_{2},\ldots,y_{n}=x_{n}$, so we get a coordinate system $(y_{1},\ldots,y_{n})$ with $\Omega=d(y_{1}^{2}/2)\wedge dy_{2}\wedge\ldots\wedge dy_{n}=y_{1}dy_{1}\wedge\ldots\wedge dy_{n}$. (In the real case with $h(O)<0$, put $y_{1}=x_{1}\sqrt{-2h},y_{2}=-x_{2},y_{3}=x_{3},\ldots,y_{n}=x_{n}$.) ∎

Let $X$ be a vector field which preserves a folded volume form $\Omega$. If $X(O)\neq 0$ then it is easy to see that $X$ can be rectified together with $\Omega$, i.e., there is a coordinate system $(y_{1},\ldots,y_{n})$ in which

[TABLE]

Indeed, consider the local $(n-1)$-dimensional quotient space of the local flow generated by $X$. The local $(n-1)$-form $X\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\Omega$ is the pull-back of a local folded volume form on this local quotient space so we can write $X\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\Omega=y_{1}dy_{1}\wedge\ldots\wedge dy_{n-1}$ by the previous Lemma 3.1. Now take any function $y_{n}$ such that $X(y_{n})=1$ and $y_{n}(O)=0$ and we are done.

Consider now the case when $X(O)=0$. We have the following theorem, which is a generalization of the results about normalization of isochore vector fields (see [37, Section 4]) from the case of a regular volume form to the case of a folded volume form.

Theorem 3.2**.**

Let $X$ be a formal vector field which vanishes at a point $O$ and which preserves a formal folded volume form $\Omega$. Then $X$ can be formally normalized together with $\Omega$: $\Omega$ has canonical form and $X$ is in normal form in the coordinate system $(x_{1},\ldots,x_{n})$ and the semisimple part of $X$ is diagonal in this system (over $\mathbb{C}$):

[TABLE]

The eigenvalues of $X$ satisfy the following resonance relation:

[TABLE]

Moreover, if $X$ and $\Omega$ are analytic and $X$ is analytically integrable or Darboux integrable, then there exists such a normalization which is locally analytic.

Proof.

By Lemma 3.1 we may write $\Omega=y_{1}dy_{1}\wedge\ldots\wedge dy_{n}$ in some coordinate system. Denote by $\rho$ the associated torus $\mathbb{T}^{\tau}$-action of $X$ at $O$, where $\tau$ is the toric degree of $X$ at $O$. Theorem 2.3 ensures that $\rho$ preserves $\Omega$. This implies that the singular set $S=\{y_{1}=0\}$ of $\Omega$ is a hyperplane which is preserved by $\rho$. Since $\Omega$ is homogeneous in the coordinate system $(y_{1},\ldots,y_{n})$, the linear part of the action $\rho$ in this coordinate system, which is a linear torus action denoted by $\rho_{1}$, also preserves $\Omega$. Notice that, since $S=\{y_{1}=0\}$ is a hyperplane preserved by $\rho$, it is also preserved by $\rho_{1}$.

By Bochner’s averaging formula [5], we find a formal diffeomorphism $\Phi$, whose linear part is the identity, and which intertwines $\rho$ with $\rho_{1}$, i.e.,

[TABLE]

Let

[TABLE]

Then $\rho_{1}$ also preserves $\Omega_{1}$: $\rho_{1}(s,.)^{*}\Omega_{1}=\rho_{1}(s,.)^{*}\Phi^{*}\Omega=(\Phi\circ\rho_{1}(s,.))^{*}\Omega=(\rho(s,.)\circ\Phi)^{*}\Omega=\Phi^{*}\rho(s,.)^{*}\Omega=\Phi^{*}\Omega=\Omega_{1}$ for any $s\in\mathbb{T}^{\tau}.$ Thus $\rho_{1}$ preserves both $\Omega$ and $\Omega_{1}$.

Since $S=\{y_{1}=0\}$ is preserved by $\rho$, it is also preserved by $\Phi$, which implies that it is also the singular set of $\Omega_{1}$, i.e., $\Omega_{1}=\Phi^{*}\Omega$ is also divisible by $y_{1}$. Moreover, the linear part of $\Phi$ is the identity. So we can write

[TABLE]

where $\Theta$ is a differential $n$-form which vanishes at $O$.

The next step is to use the equivariant Moser path method to move $\Omega_{1}$ to $\Omega$ by a formal diffeomorphism without changing the torus action. According to Lemma 2.5, it is enough to show that the equation

[TABLE]

has a solution $Y_{t}$ (for $t\in[0,1]$).

We can write $y_{1}\Theta=d(y_{1}\Pi)$ for some $(n-1)$-form $\Pi$ with $\Pi(O)=0$ and solve the following equation instead:

[TABLE]

or, equivalently,

[TABLE]

The equation above clearly admits a unique solution $Y_{t}$ because $dy_{1}\wedge\ldots\wedge dy_{n}$ is regular and $\Theta(O)=0$ and, moreover, $Y_{t}(O)=0$ because $\Pi(O)=0$.

Thus we have shown the existence of a simultaneous normalization of $X$ and $\Omega$, i.e., we found a coordinate system $(y_{1},\ldots,y_{n})$ in which $X^{S}=X^{s}$ (the semisimple part of $X$ coincides with the semisimple part of its linear part) and $\Omega=y_{1}dy_{1}\wedge dy_{2}\wedge\ldots\wedge dy_{n}$. A priori, $X^{s}$ is not diagonal in the coordinates $(y_{1},\ldots,y_{n})$, so we have a bit more work to do to diagonalize it without destroying the form of $\Omega.$

Recall that the $(n-1)$-dimensional singular set $S=\{y_{1}=0\}$ of $\Omega$ is invariant under the action $\rho$, so $\rho$ also preserves a line $L$ transversal to $S$. This line can be parametrized by $y_{1}$:

[TABLE]

where $a_{2},\ldots,a_{n}\in\mathbb{C}$ are constants. By the linear transformation

[TABLE]

we get a new coordinate system in which $\rho$ still linear, $\Omega$ still has the same form, and both $S=\{z_{1}=0\}$ and $L=\{z_{2}=\ldots=z_{n}=0\}$ are invariant with respect to $\rho$. This means that $\rho=\rho_{S}\oplus\rho_{L}$ where $\rho_{S}$ is a linear $\mathbb{T}^{\tau}$-action on $S$ and $\rho_{L}$ is a linear $\mathbb{T}^{\tau}$-action on $L$. Since $\rho$ is determined by $X^{S}$, we have a corresponding decomposition $X^{S}=X^{S}_{S}\oplus X^{S}_{L}$ for $X^{S}$. Since $\Omega$ also decomposes as $\Omega=\Omega_{S}+\Omega_{L}$ with $\Omega_{S}=z_{1}dz_{1}$ and $\Omega_{L}=dz_{2}\wedge\ldots\wedge dz_{n}$, the fact that $X^{S}$ preserves $\Omega$ implies that $X^{S}_{S}$ preserves $\Omega_{S}$ and $X^{S}_{L}$ preserves $\Omega_{L}$. From this it is easy to see that we can diagonalize $X^{S}_{S}$ on $S$ without destroying the form of $S$, i.e., $\Omega_{S}=dx_{2}\wedge\ldots\wedge dx_{n}$ and $X^{S}_{S}=\sum_{i=2}^{n}\gamma_{i}x_{i}\dfrac{\partial}{\partial x_{i}}.$ Since $L$ is only 1-dimensional, $X^{S}_{L}$ is already diagonal, so we can put $x_{1}=z_{1}$. Then in the coordinate system $(x_{1},\ldots,x_{n})$ we have that $X$ is diagonal and $\Omega$ still has the required canonical form.

This normalization is, a priori, only formal. However, if $\Omega$ is analytic and $X$ is analytically or Darboux integrable, the associated torus action $\rho$ is analytic and all the steps above can be done analytically, so we have a local analytic normalization. The resonance equation (3.1) is the same as the equation $\mathcal{L}_{X^{S}}\Omega=0$ in diagonalized normalized coordinates. ∎

3.2. Systems preserving a singular Nambu structure of top order

We start by recalling the following result, which is a simple particular case of the linearization result of Nambu structures (see, e.g., [10]):

Consider a formal (resp. local analytic, resp. smooth) $n$-vector field

[TABLE]

in some coordinate system $(x_{1},\ldots,x_{n})$ around a point $O$ on an $n$-dimensional manifold. Such a multi-vector field of top order is also called a Nambu structure of top order. We say that $O$ is a non-degenerate singular point of $\Lambda$ if $g(O)=0$ and $dg(0)\neq 0$. Clearly, this definition does not depend on the choice of the coordinate system at $O$.

Lemma 3.3**.**

Let $\Lambda$ be a formal (resp. analytic, resp. smooth) Nambu structure of top order in dimension $n\geq 2$ with a non-degenerate singular point $O$. Then there exists a formal (resp. analytic, resp. smooth) coordinate system $(x_{1},\ldots,x_{n})$ around $O$ in which we have

[TABLE]

Consider now a vector field $X$ which preserves $\Lambda=x_{1}\dfrac{\partial}{\partial x_{1}}\wedge\dfrac{\partial}{\partial x_{2}}\wedge\ldots\wedge\dfrac{\partial}{\partial x_{n}}$ near $O$. First, we prove the following simple result if $X(O)\neq 0$.

Proposition 3.4**.**

If $X(O)\neq 0$ then there is a local coordinate system in which $X$ can be rectified together with $\Lambda$:

[TABLE]

Proof.

Since $X(O)\neq 0,$ we can rectify $X$ as $X=\dfrac{\partial}{\partial x_{n}}$ in a coordinate system $(z_{1},\ldots,z_{n-1},x_{n})$. The invariance of $\Lambda$ with respect to $X$ means that it has the form $\Lambda=f(z_{1},\ldots,z_{n-1})\dfrac{\partial}{\partial z_{1}}\wedge\ldots\wedge\dfrac{\partial}{\partial z_{n-1}}\wedge X=\Pi\wedge X$, where $\Pi=f(z_{1},\ldots,z_{n-1})\dfrac{\partial}{\partial z_{1}}\wedge\ldots\wedge\dfrac{\partial}{\partial z_{n-1}}$. By Lemma 3.3, we can write $\Pi=x_{1}\dfrac{\partial}{\partial x_{1}}\wedge\ldots\wedge\dfrac{\partial}{\partial x_{n-1}}$ by a change of coordinates from $(z_{1},\ldots,z_{n-1})$ to $(x_{1},\ldots,x_{n-1})$ which does not involve $x_{n}$. Then (3.3) is satisfied in the new coordinate system $(x_{1},\ldots,x_{n-1},x)$. ∎

Consider now the case when $X(O)=0$.

Theorem 3.5**.**

Let $X$ be a formal vector field which vanishes at the origin $O$ and which preserves a Nambu structure of top order $\Lambda$ with a non-degenerate singularity at $O$. Then $X$ can be formally normalized together with $\Lambda$: there is a formal coordinate system $(x_{1},\ldots,x_{n})$ in which $\Lambda$ has canonical form, $X$ is in normal form, and the semisimple part of $X$ is diagonal in this coordinate system (over $\mathbb{C}$):

[TABLE]

The eigenvalues of $X$ satisfy the following resonance relation:

[TABLE]

Moreover, if $X$ and $\Lambda$ are analytic and $X$ is analytically or Darboux integrable, then there exists such a normalization which is locally analytic.

Proof.

The proof is quite similar to the case of a folded volume form. We start with a coordinate system in which $\Lambda$ is in canonical form (3.2) but the associated torus action $\rho$ of $X$ is, a priori, nonlinear. Since $\Lambda$ is homogeneous and $\rho$ preserves $\Lambda$, the linear part $\rho_{1}$ of $\rho$, which is a linear torus action, also preserves $\Lambda$. By Bochner’s averaging formula, we find a formal diffeomorphism $\Psi$ whose linear part is the identity and which intertwines $\rho$ with $\rho_{1}$. Then $\rho_{1}$ preserves both $\Lambda$ and $\Lambda_{1}=\Psi_{*}\Lambda$. Similar to the case of a folded volume form, $\Lambda$ and $\Lambda_{1}$ have the same singular set $S$, which is a linear subspace of codimension 1 in some coordinate system in which $\Lambda$ is linear and the linear part of $\Lambda_{1}$ coincides with $\Lambda$, i.e., we can write

[TABLE]

where $\Pi$ is an $n$-vector field which vanishes at $O$. To apply Lemma 2.5, we must find a time-dependent vector field $Y_{t}$ such that

[TABLE]

In order to simplify the equation above, we can impose $Y_{t}(x_{1})=0$ (i.e., $x_{1}$ is a first integral for $Y_{t}$) and define $\Omega_{t}=\left(\dfrac{\partial}{\partial x_{1}}\wedge\ldots\wedge\dfrac{\partial}{\partial x_{n}}+t\Pi\right)^{-1}$, which is a regular volume form since $P(O)=0$. Put $g_{t}=\langle\Pi,\Omega_{t}\rangle$. Then the above equation, after pairing both sides by $-\dfrac{\Omega_{t}\langle\Omega_{t},\cdot\rangle}{x_{1}}$, becomes

[TABLE]

which can be easily solved. The rest of the proof is similar to that of Theorem 3.2. ∎

4. Systems with a singular symplectic structure

In this section we consider two kinds of singular symplectic structures, namely the so called folded symplectic structures (which have vanishing components) and log-symplectic structures (which have poles). We denote by $\mathbb{K}$ either $\mathbb{R}$ or $\mathbb{C}$.

4.1. Folded symplectic structures

Consider a (formal, or local analytic, or smooth) closed 2-form $\omega$ on $(\mathbb{K}^{2n},O)$, such that $\omega^{n}$ is non-trivial on $\mathbb{K}^{2n}$ but $\omega^{n}(O)=0$. Assume that the corank of $\omega$ at $O$ is $2$, and that $\omega^{n}=fdx_{1}\wedge\ldots\wedge dx_{2n}$ in some local coordinate system, where $f(O)=0$ but $df(O)\neq 0$, so that $S=\{f=0\}$ is the hypersurface of singular points of $\omega$. Assume, moreover, that the pull-back of $\omega$ to $S$ has constant rank equal to $2(n-1)$. Such a closed 2-form $\omega$ is called a folded symplectic structure (because it can be obtained as the pull-back of a usual symplectic form by a fold map [21]); it is known to admit the following (formal, or local analytic, or smooth) normal form, according to a classical result of Martinet [20]:

[TABLE]

More generally, one may consider a multi-folded symplectic structure $\omega$, which for some positive number $m\leq n$ admits the following canonical Darboux-like expression in some coordinate system $(x_{1},\ldots,x_{2m},y_{1},\ldots,y_{2(n-m)})$:

[TABLE]

The multi-folded symplectic structure above is quasi-homogeneous: we have

[TABLE]

where $E$ is the quasi-Euler vector field

[TABLE]

In other words, if we declare that each $x_{i}$ is of quasi-homogeneous order $2$ and each $y_{j}$ is of quasi-homogeneous order $3$, then $\omega$ becomes quasi-homogeneous of order 6.

Notice that the kernel of $\omega$ at $O$ is the $2m$-dimensional vector space

[TABLE]

The quasi-Euler vector field $E$ is tangent to $K$ and is equal to 2 times the usual Euler vector field on $K$. $E$ also projects to 3 times the usual Euler vector field on the quotient space $\mathbb{K}^{2n}/K$.

Let us assume now that there is a local (smooth, analytic, or formal) action $\rho$ of a compact Lie group $G$ on $(\mathbb{K}^{2n},O)$ which fixes the origin $O$ and preserves $\omega$. A priori, $\rho$ does not preserve the quasi-Euler vector field $E$, but it must preserve the kernel space $K$ because it preserves $\omega$. Denote by

[TABLE]

the average of $E$ with respect to $\rho$ (where $d\mu_{G}$ denotes the Haar measure on $G$).

The vector field $E_{\rho}$ is preserved by $\rho$ and we still have $\mathcal{L}_{E_{\rho}}\omega=6\omega$. Moreover, $E_{\rho}$ is still tangent to $K$ and has its linear part equal to 2 times the usual Euler vector field on $K$; the projection of the linear part of $E_{\rho}$ on the quotient space $\mathbb{K}^{2n}/K$ is still equal to 3 times the usual Euler vector field on this quotient space. All of this implies, in particular, that the linear part $E_{\rho}^{(1)}$ of $E_{\rho}$ is, a priori, non-diagonal but diagonalizable, with only two distinct eigenvalues 2 and 3: to diagonalize $E_{\rho}^{(1)}$ we must apply a linear transformation to $\mathbb{K}^{2n}$, which is, a priori, not identity, but which is the identity on $K$ and also projects to the identity map on $\mathbb{K}^{2n}/K$. In other words, $E_{\rho}^{(1)}$ is diagonal in a coordinate system $(x_{i}^{\prime},y_{j}^{\prime})$ of the type

[TABLE]

for some constants $a_{ij}$.

Notice that such a vector field is in the Poincaré domain and is non-resonant in the sense of Poincaré-Dulac, i.e., there is no resonance relation of the type $\gamma_{i}=\gamma_{j_{1}}+\dots+\gamma_{j_{k}}$ (with $k\geq 2$) among its eigenvalues. It follows from the classical theory of normalization of vector fields that $E_{\rho}$ is locally diagonalizable (smoothly if $E_{\rho}$ and $\rho$ are smooth, analytically if $E_{\rho}$ and $\rho$ are analytic, and formally if $E_{\rho}$ and $\rho$ are formal), also in an equivariant way. (For the smooth case, see [3]; the formal and analytic cases are simpler and can be done using the toric approach.) In other words, there is another coordinate system $(u_{1},\ldots,u_{2m},v_{1},\ldots,v_{2(n-m)})$ on $(\mathbb{K}^{2n},O)$, such that $u_{i}=x_{i}^{\prime}+h.o.t.$ and $v_{i}=y_{i}^{\prime}+h.o.t.$ ($h.o.t.$ means higher order terms, i.e., terms of degree at least 2 here), in which the action $\rho$ is linear, and the vector field $E_{\rho}$ has the following diagonal form:

[TABLE]

Since the linear action $\rho$ (in the coordinates $(u_{i},v_{j})$) preserves $E_{\rho}$, it must also preserve the eigenspaces $U$ and $V$ of $E_{\rho}$, where $U$ is the space where all the coordinates $v_{i}$ vanish and $V$ is the space where all the coordinates $u_{i}$ vanish, and we have the splitting $\mathbb{K}^{2n}=U\oplus V$.

Remember that $\mathcal{L}_{E_{\rho}}(\omega)=6\omega$. Now, $E_{\rho}$ is diagonal, and the monomial 2-forms are eigenvectors of the Lie derivative operator $\mathcal{L}_{E_{\rho}}$. The only monomial 2-forms which have eigenvalue equal to 6 are $dv_{i}\wedge dv_{j}$ and $u_{k}du_{i}\wedge du_{j}$, so $\omega$ must be a linear combination with constant coefficients of these monomial forms. (By standard division techniques, one can show easily that this is also true in the smooth case). It follows that $\omega$ admits an equivariant splitting

[TABLE]

where $\omega_{V}$ is defined on $V$ and is constant, $\omega_{U}$ is defined on $U$ and is linear, and $\rho$ preserves both $\omega_{U}$ and $\omega_{V}$ (because it acts separately on $U$ and $V$ and cannot mix up these terms).

Recall that, by construction, we have

[TABLE]

and

[TABLE]

for every $i\leq 2m$ and every $j\leq 2(n-m)$. Putting these into the original formula (4.1) of $\omega$, we get

[TABLE]

In the expression above, the sum of linear $U$-terms (i.e., linear terms that do not contain any $v_{j}$ or $dv_{j}$ in their expression) is $\sum_{i=1}^{m}u_{i}du_{i}\wedge u_{m+i}$. Since $\omega=\omega_{U}+\omega_{V},$ with $\omega_{U}$ being a sum of linear $U$-terms, we automatically have $\omega_{U}=\sum_{i=1}^{m}u_{i}du_{i}\wedge u_{m+i}$. By a similar argument, we also automatically have $\omega_{V}=\sum_{j=1}^{n-m}dv_{j}\wedge dv_{j+n-m}.$ Thus we have proved the following equivariant normalization theorem for multi-folded symplectic structures.

Theorem 4.1**.**

Consider a multi-folded symplectic structure $\omega$ written as

[TABLE]

($1\leq m\leq n$) in some local (analytic, resp. formal, resp. smooth) coordinate system $(x_{1},\ldots,x_{2m},y_{1},\ldots,y_{2(n-m)})$ on $(\mathbb{K}^{2n},O)$. Assume that there is a local (analytic, resp. formal, resp. smooth) action $\rho$ of a compact Lie group $G$ on $(\mathbb{K}^{2n},O)$ which fixes the origin $O$ and preserves $\omega$. Then there exists a local (analytic, resp. formal, resp. smooth) coordinate system $(u_{1},\ldots,u_{2m},v_{1},\ldots,v_{2(n-m)})$ on $(\mathbb{K}^{2n},O)$, in which the action $\rho$ is linear and the multi-folded symplectic form $\omega$ is still in canonical form:

[TABLE]

As an immediate consequence of the theorem above, we get the following result about normalization of a vector field with an underlying (multi-)folded symplectic structure.

Theorem 4.2**.**

Let $X$ be a formal vector field on $(\mathbb{K}^{2n},O)$ which vanishes at $O$ and preserves a (multi-)folded symplectic structure $\omega$ with a canonical expression $\omega=\sum_{i=1}^{m}x_{i}dx_{i}\wedge dx_{m+i}+\sum_{j=1}^{n-m}dy_{j}\wedge dy_{j+n-m}$ in some coordinate system. Then the couple $(X,\omega)$ admits a formal simultaneous normalization, i.e., there is a formal coordinate transformation which normalizes $X$ while keeping the canonical expression of $\omega$. Moreover, if everything is locally analytic (real or complex), and $X$ is analytically or Darboux integrable, then this normalization can be made locally analytic.

Proof.

Just apply Theorem 4.1 to the associated torus action of $X$ at $O$. ∎

4.2. Log-symplectic structures

Recall that a meromorphic differential form $\omega$ on a complex manifold $M$, with poles along a divisor $D\subset M$ (without multiplicities), is called logarithmic if both forms $\omega$ and $d\omega$ have poles only along the divisor $D$ and of order not greater than 1. Such logarithmic differential forms have been studied by many people in algebraic geometry, see, e.g., Deligne [8], Saito [30], Aleksandrov [2]. Roughly speaking, the logarithmic condition means that if $\omega$ contains a local pole $1/h$, where $h$ is a local holomorphic function whose zero locus is a component of $D$ (without multiplicities), then this pole must come together with $dh$, i.e., one can write a local expression of $\omega$ which contains only holomorphic terms and meromorphic terms of the type $dh/h=d(\log h)$; otherwise $d\omega$ would contain a pole of order 2 at the zero locus of $h$.

Consider a local logarithmic differential 2-form $\omega$. Then it admits the following expression:

[TABLE]

where $g_{ij},\beta_{k},\gamma$ are local holomorphic functions, 1-forms, and 2-forms, respectively. The general case with non-trivial terms of the type $g_{ij}\dfrac{dh_{i}}{h_{i}}\wedge\dfrac{dh_{j}}{h_{j}}$ is very interesting but also complicated, so here we will be less ambitious and study only local logarithmic 2-forms without such terms. We introduce the following definition.

Definition 4.3**.**

A local differential 2-form of the type

[TABLE]

on $(\mathbb{K}^{2n},O)$ is called a simple log-symplectic form if it satisfies the following conditions:

i) The local functions $h_{1},\ldots,h_{k}$ vanish at the origin $O$, but $dh_{1}\wedge\ldots\wedge dh_{k}(O)\neq 0.$

ii) $d\omega=0$.

iii) $h\omega^{n}$ is a local regular volume form, i.e., $h\omega^{n}(O)\neq 0,$ where $h=\prod_{i=1}^{k}h_{i}$ and $\omega^{n}$ means the wedge product of $n$ copies of $\omega$.

The definition above is valid in many different categories: real analytic, holomorphic, formal, and smooth. Log-symplectic manifolds, especially in the case with $k=1$ (i.e., the set of singular points is a smooth hypersurface), have been studied by many authors from different points of view, where they may be called by other names as well, such as $b$-symplectic or $b$-Poisson (the case with $k=1$), or $c$-symplectic, see, e.g., [14, 12, 13].

It turns out that non-degenerate log-symplectic structures admit Darboux-like local normal forms.

Theorem 4.4**.**

Let $\omega$ be a local (analytic, formal, or smooth) simple log-symplectic structure in $(\mathbb{K}^{2n},O)$ whose divisor has $k$ components. Then there exists a local (analytic, formal or smooth) coordinate system $(x_{1},y_{1},\ldots,x_{k},y_{k},z_{1},\ldots,z_{2(n-k)})$ in which $\omega$ has the following canonical expression:

[TABLE]

In particular, the dual Poisson structure $\Pi$ of $\omega$ is without poles and admits the following canonical form:

[TABLE]

Proof.

Our proof is based on the path method and consists of four small steps.

Step 1. We begin with a log-symplectic form $\omega$ written as

[TABLE]

where $\beta_{i}$ are regular 1-forms and $\gamma$ is a regular 2-form. The closedness condition on $\omega$ gives

[TABLE]

which implies that, for each $i\leq k$, $dh_{i}\wedge d\beta_{i}$ is divisible by $h_{i}$. In turn, this easily implies, by Poincaré’s Lemma, that $\beta_{i}$ can be written as $\beta_{i}=h\beta_{i}^{\prime}+gdh_{i}+dy_{i}$. Indeed, we can write $\beta_{i}=h_{i}\beta_{i}^{\prime}+gdh_{i}+\xi_{i}$ in a coordinate system $(h_{i},v_{1},\ldots,v_{2n-1})$, in which $\xi_{i}$ is the part of $\beta_{i}$ that is independent of $h_{i}$ and does not contain $dh_{i}$. The condition that $dh_{i}\wedge d\beta_{i}$ is divisible by $h_{i}$ means that $dh_{i}\wedge d\xi_{i}$ is divisible by $h_{i}$, which then implies that $d\xi_{i}=0$ (otherwise $dh_{i}\wedge d\xi_{i}$ would be non-trivial and not divisible by $h_{i}$). So, by Poincaré’s Lemma, we can write $\xi_{i}=dy_{i}$. We can forget the term $gdh_{i}$ in the expression of $\beta_{i}$ because its wedge product with $\dfrac{dh_{i}}{h_{i}}$ is zero. Hence we can assume that $\beta_{i}=h_{i}\beta_{i}^{\prime}+dy_{i}$. Notice that $\dfrac{dh_{i}}{h_{i}}\wedge h\beta_{i}$ is regular and can be added to $\gamma$, so we have

[TABLE]

where $y_{i}$ are local functions (formal, analytic, or smooth) and $\mu$ is a 2-form.

The value of $h\omega^{n}$ at $O$, where $h=\prod_{i=1}^{k}h_{i}$ and $\omega^{n}$ means the wedge product of $n$ copies of $\omega$, is equal to

[TABLE]

if $k\leq n$, and is equal to 0 if $k>n$. This implies that $k\leq n$ and $\wedge_{i=1}^{k}(dh_{i}\wedge dy_{i})\wedge\mu^{n-k}(O)$ is a regular volume form. In particular, $(h_{1},y_{1},\ldots,h_{k},y_{k})$ can be completed to a regular local coordinate system $(h_{1},y_{1},\ldots,h_{k},y_{k},z_{1},\ldots,z_{2n-2k})$.

Step 2. Write $\mu$ as

[TABLE]

where $\phi_{i}$ and $\psi_{i}$ are linear functions in the coordinates $(h_{1},y_{1},\ldots,h_{k},y_{k},z_{1},\ldots,z_{2n-2k})$, $\mu_{0}$ is a constant 2-form in these coordinates which contains only terms $dz_{i}\wedge dz_{j}$, and $\mu_{1}$ is a 2-form which vanishes at the origin. Then $\omega$ can be written as

[TABLE]

where $\mu_{2}$ is a 2-form which vanishes at $O$. In other words, we have

[TABLE]

where $\tilde{h}_{i}=h_{i}\exp(\psi_{i})$ and $\tilde{y_{i}}=y_{i}+h_{i}\phi_{i}$. Notice that $\tilde{h}_{i}=h_{i}+h.o.t.$ (higher order terms) and $\tilde{y}_{i}=y_{i}+h.o.t.$, so $(\tilde{h}_{1},\tilde{y}_{1},\ldots,\tilde{h}_{k},\tilde{y}_{k},z_{1},\ldots,z_{2n-2k})$ is still a local coordinate system. The condition $h\omega^{n}(O)\neq 0$ means that $\mu_{0}^{n-k}(O)\neq 0$, i.e., $\mu_{0}^{n-k}(O)$ is a symplectic form of degree $2(n-k)$ in the coordinates $(z_{1},\ldots,z_{2n-2k})$. By the Darboux theorem, by a linear change of the coordinates $(z_{1},\ldots,z_{2(n-k)})$, we may assume that

[TABLE]

Step 3. Put

[TABLE]

and define the following linear path of 2-forms for $t\in[0,1]$:

[TABLE]

where $h_{i,t}=h_{i}\exp(t\psi_{i})$, $y_{i,t}=y_{i}+th_{i}\phi_{i}$, and $\mu_{1,t}$ is a 2-form depending polynomially on $t$ which vanishes at the origin.

Due to the closedness of $\omega$, we can also write

[TABLE]

where $\xi$ is an 1-form which vanishes at the origin.

To show that $\omega$ is locally (or formally) isomorphic to the canonical form $\omega_{0}$ via a local (or formal) diffeomorphisms, we now invoke the path method with respect to the path $(\omega_{t})$ and construct the time-1 flow of the time-dependent vector field $X_{t}$ given by the (time-dependent) equation:

[TABLE]

Step 4. The last step is to verify that the equation above does admit a local (formal, analytic, or smooth) solution $X_{t}$ (which depends smoothly on $t$) such that $X_{t}(O)=0.$ Indeed, the 2-form

[TABLE]

sends the vector fields $h_{i,t}\dfrac{\partial}{\partial h_{i,t}}$, $h_{i,t}\dfrac{\partial}{\partial y_{i,t}}$, $\dfrac{\partial}{\partial z_{j}}$, $\dfrac{\partial}{\partial z_{j+n-k}}$ (written with respect to the coordinate system $(h_{i,t},y_{i,t},z_{j})$), via contraction, to the 1-forms (up to a sign) $dy_{i,t}$, $dh_{i,t}$, $dz_{j+n-k}$, $dz_{j}$. The contraction map is hence an isomorphism from the submodule of vector fields generated by the vector fields $\left\{h_{i,t}\dfrac{\partial}{\partial h_{i,t}},\,y_{i,t}\dfrac{\partial}{\partial y_{i,t}},\,\dfrac{\partial}{\partial z_{j}},\,\dfrac{\partial}{\partial z_{j+n-k}}\right\}$ to the module of 1-forms. Since $\mu_{1,t}$ is a small perturbation term (because it vanishes at $O$), the contraction map by $\omega_{t}$ is also an isomorphism between these two modules, so the equation $X_{t}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega_{t}=\xi$ can be solved (with an unique solution) and, moreover, $X_{t}(O)=0$ because $\xi(O)=0$. This finishes the proof. (The $h_{i}$ are renamed $x_{i}$ at the end.) ∎

Remark 4.5*.*

The case with $k=1$ of Theorem 4.4 was obtained before by Gulliemin-Miranda-Pires [14], though the general case (with arbitrary $k$) does not seem to have been written down explicitly anywhere before, as far as we know. One can also prove Theorem 4.4 differently, e.g., by the following steps: i) Show that the associated Poisson structure $\Pi$ of $\omega$ is smooth; ii) Show that the Lie algebra which corresponds to the linear transversal part of $\Pi$ at the origin is isomorphic to the direct sum of $k$ copies of $af\!f(1)$, where $af\!f(1)$ denotes the 2-dimensional Lie algebra of affine transformations of the line; iii) Use the result of Dufour and Molinier [9] about the linearizability of Poisson structures with such a linear part. Our proof of Theorem 4.4 is written in a way that immediately extends to the equivariant case when there is a compact group action which preserves the log-symplectic form.

4.3. Equivariant normalization of log-symplectic structures

Theorem 4.6**.**

Let $\omega$ be a local (formal, analytic, or smooth) non-degenerate log-symplectic structure in $(\mathbb{K}^{2n},O)$ whose divisor contains $k$ components. Let $\rho$ be a local (formal, analytic, or smooth) action of a compact Lie group $G$ on $(\mathbb{K}^{2n},O)$ which fixes the origin $O$ and which preserves $\omega$. Then there exists a local (formal, analytic, or smooth) coordinate system $(x_{1},y_{1},\ldots,x_{k},y_{k},z_{1},\ldots,z_{2(n-k)})$ in which the action $\rho$ is linear and the form $\omega$ has the following Darboux-like canonical expression:

[TABLE]

Proof.

By Theorem 4.4, we already know that, if we forget about the compact Lie group action, then there is a local coordinate system in which the dual Poisson structure $\Pi$ of $\omega$ has the expression

[TABLE]

Using this explicit formula, one verifies easily that $\Pi$ satisfies the following division property: any local (formal, analytic, or smooth) vector field $X$ which is tangent to the symplectic leaves of $\Pi$ is a linear combination, with coefficients which are local functions, of local Hamiltonian vector fields of $\Pi$. A result of Miranda and Zung [24] then says that $\Pi$ admits an equivariant splitting. In fact, even without this division property, Poisson structures would still admit an equivariant splitting, according to a more recent result by Frejlich and Mărcuţ [11]. This means that one can split $\Pi$ equivariantly with respect to the $G$-action into a direct sum of two parts: the symplectic part and the part which vanishes at zero (like in Weinstein’s splitting theorem [34]); the $G$-action is diagonal, acting on each factor separately. So the equivariant normalization of $\Pi$ amounts to an equivariant normalization of two separate parts: the symplectic part (given by the equivariant Darboux theorem), and the part which vanishes at zero. Thus, this equivariant splitting result reduces the proof of Theorem 4.6 to the case when the rank of $\Pi$ at $O$ is zero, i.e., the case when $k=n$.

Now we can assume, by the case $k=n$ of Theorem 4.4, that $\omega$ is of the form

[TABLE]

in some local (or formal) coordinate system $(x_{i},y_{i})$ (in which the action $\rho$ of $G$ is, a priori, non-linear). We want to linearize the action of $G$ while keeping the above canonical expression of $\omega$. Notice that, due to the homogeneity of $\omega$, the linear part of $\rho$ (with respect to the coordinate system $(x_{i},y_{i})$), denoted by $\rho_{1}$, is a linear $G$-action which also preserves $\omega$. By Bochner’s local linearization theorem for compact group actions [5], we know that $\rho$ is isomorphic to $\rho_{1}$ via a local (or formal) diffeomorphism $\Phi$ whose linear part is the identity. This means that the couple $(\omega,\rho)$ is isomorphic to the couple $(\omega^{\prime},\rho_{1})$ via $\Phi$, where $\omega^{\prime}=\Phi_{*}\omega$. Due to the fact that $\rho$ must preserve the hyperplanes $S_{i}=\{x_{i}=0\}$, we have that $\Phi_{*}x_{i}$ (obtained by Bochner’s averaging formula) is divisible by $x_{i}$. This implies that $\omega^{\prime}$ has the form

[TABLE]

where $\mu$ is a local holomorphic (or formal, or smooth) 2-form. Now we can simply repeat the steps of the proof of Theorem 4.4, but in a $\rho_{1}$-equivariant way, to conclude that $(\omega^{\prime},\rho_{1})$ is isomorphic to $(\omega,\rho_{1})$. Thus, $(\omega,\rho)$ is isomorphic to $(\omega^{\prime},\rho_{1})$, which is isomorphic to $(\omega,\rho_{1})$. The composed diffeomorphism which moves $(\omega,\rho)$ to $(\omega,\rho_{1})$ is our required normalization map. ∎

As an immediate consequence of Theorem 4.6, we obtain the following result.

Theorem 4.7**.**

Let $X$ be a formal vector field on $(\mathbb{K}^{2n},O)$ which vanishes at $O$ and preserves a non-degenerate log-symplectic symplectic structure $\omega$ with a canonical expression $\omega=\sum_{i=1}^{k}\dfrac{dx_{i}}{x_{i}}\wedge dy_{i}+\sum_{j=1}^{n-k}dz_{j}\wedge dz_{j+n-k}$ in some coordinate system. Then the couple $(X,\omega)$ admits a formal simultaneous normalization, i.e., there is a formal coordinate transformation which normalizes $X$ while keeping the canonical expression of $\omega$. Moreover, if everything is locally analytic (real or complex), and $X$ is analytically or Darboux integrable, then this normalization can be made locally analytic.

Proof.

Just apply Theorem 4.6 to the associated torus action of $X$ at $O$. ∎

5. Systems with a singular contact structure

In this section, we study generic singular contact distributions $\xi$ which are given by the kernel of singular contact forms $\alpha=\sum_{i=0}^{2n}f_{i}(x)dx_{i}$ on $(\mathbb{K}^{2n+1},O)$, where $O$ denotes the origin of $\mathbb{K}^{2n+1}$. The word “generic” here means that $d\alpha$ is a regular presymplectic form of rank $2n$ near $O$, i.e., $(d\alpha)^{n}(O)\neq 0$, and that, if we write $\alpha\wedge(d\alpha)^{n}=fdx_{0}\wedge dx_{1}\wedge\cdots\wedge dx_{2n}$ for some function $f$, then $f(O)=0$ but $df(O)\neq 0$.

We consider the following two types of singular contact forms $\alpha=\sum_{i=0}^{2n}f_{i}(x)dx_{i}$:

1. (1)

If $\alpha(O)\neq 0$, i.e., the kernel distribution $\xi=\ker\alpha$ is still regular at $O$, then $\alpha$ is called a non-vanishing singular contact form. 2. (2)

Suppose that $\alpha(O)=0$ and, in addition, it satisfies the following non-degeneracy condition:

[TABLE]

Then $\alpha$ is called a non-degenerate singular contact form.

Definition 5.1**.**

Let $X$ be a vector field, $\phi_{X}^{t}$ its local flow, and $\xi$ a singular contact distribution defined by some singular contact form $\alpha$ on a $(2n+1)$-dimensional manifold $M$. We say that $X$ preserves $\xi$ if the pull back $(\phi_{X}^{t})^{*}\alpha=f_{t}\alpha$, where $f_{t}$ is a family of smooth functions on $M$. In particular, we say the vector field preserves the singular contact form $\alpha$ if $f_{t}\equiv 1$.

It is easy to see that $X$ preserves $\xi$ if and only if $\alpha$ is a semi-invariant of $X$, i.e., $\mathcal{L}_{X}\alpha=\left(\left.\dfrac{d}{dt}\right|_{t=0}f_{t}\right)\alpha$. The vector field $X$ preserves $\alpha$ if and only if $\mathcal{L}_{X}\alpha=0$.

If a vector field $X$ preserves a singular contact form $\alpha$, then it naturally preserves the hypersurface $S=\{x\ |\ \alpha\wedge(d\alpha)^{n}(x)=0\}$ of singular points of $\alpha$, i.e., $S$ is invariant under the flow of $X$. In what follows, we focus on the relatively simpler case when $X$ preserves the singular contact form.

5.1. Non-vanishing singular contact forms

In this subsection, we assume that $\alpha(O)\neq 0$, $(d\alpha)^{n}(O)\neq 0$ and $\alpha\wedge(d\alpha)^{n}$ is a folded volume form. Then the $2n$-dimensional distribution $\xi$ is a contact distribution except on the hypersurface $S$ of singular points. A classical result of Martinet [20] states that $\alpha$ admits a Darboux normal form

[TABLE]

whose set of singular points is $S=\{\theta=0\}$ in these coordinates. In the real case, the different signs in front of the first term are important because they give different nonequivalent normal forms. Since the proofs below are identical with both signs, we shall ignore them from now on.

Lemma 5.2**.**

Suppose that $X=f_{0}\dfrac{\partial}{\partial\theta}+\sum_{i=1}^{2n}f_{i}\dfrac{\partial}{\partial x_{i}}$ preserves the singular contact form (5.1), i.e., $\mathcal{L}_{X}\alpha=0$. Then $f_{0}\equiv 0$ and $f_{1},\ldots,f_{2n}$ are independent of $\theta$.

Proof.

Since $X$ preserves $\alpha$, it also preserves the presymplectic form $\omega=d\alpha$, and hence it preserves the kernel $\mathbb{K}\dfrac{\partial}{\partial\theta}$ of $\omega$. Therefore, we have

[TABLE]

i.e., $\left[\dfrac{\partial}{\partial\theta},X\right]$ is also in the kernel of $\omega$. On the other hand, we have

[TABLE]

Therefore, $f_{1},\ldots,f_{2n}$ are independent of $\theta$.

Since $0=\mathcal{L}_{X}\alpha=X\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!d\alpha+d(X\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\alpha)$, its $d\theta$-component vanishes. Note that $X\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!d\alpha$ does not contribute any term to the $d\theta$-component, so the $d\theta$-component must come from the term $d(X\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\alpha)$. Recall that for $i=1,\ldots,n$, the coefficient function in front of $dx_{i}$ in $\alpha$, as well as $f_{i}$, are independent of $\theta$; therefore, this component is just $\dfrac{\partial(\theta f_{0})}{\partial\theta}d\theta$, which implies that $f_{0}$ is identically zero. ∎

5.1.1. The case $X(O)\neq 0$

Theorem 5.3**.**

Let $\alpha$ be a non-vanishing singular contact form on $(\mathbb{K}^{2n+1},O)$ written as $\alpha=\theta d\theta+(x_{1}+1)dx_{n+1}+\sum_{i=2}^{n}x_{i}dx_{n+i}$. Suppose $\alpha$ is preserved by a vector field $X$ with non-vanishing constant part $X^{(0)}=\sum_{i=1}^{2n}c_{i}\dfrac{\partial}{\partial x_{i}}\neq 0$. Then $X$ can be straightened out to $X=\dfrac{\partial}{\partial x_{2n}}$, if $c_{n+1}=0$, or to $X=c_{n+1}\dfrac{\partial}{\partial x_{n+1}}$, if $c_{n+1}\neq 0$, without changing the expression of $\alpha$.

Proof.

We only need to consider the $1$-form $\gamma:=\alpha-\theta d\theta$ on the hyperplane $\{\theta=0\}$ preserved by $X$ which can be regarded as a vector field on the same hyperplane by Lemma 5.2. First, we normalize the vector field $X$ by changing only the coordinates $x_{1},\ldots,x_{2n}$ and then change $\gamma$ back to normal form, keeping the expression of $X$.

If $c_{n+1}=0$, we first assume, without loss of generality, that $c_{2n}\neq 0$ and write $X=\dfrac{\partial}{\partial x_{2n}}$ after some coordinate change by a local diffeomorphism $\Phi$. Since $\gamma$ is preserved by $X$, it is independent of $x_{2n}$ in the new coordinate system and hence we can write it as a sum of a $1$-form $\gamma^{\prime}$ on $\{x_{2n}=0\}$ and a $1$-form $\gamma^{\prime\prime}=g(x_{1},\ldots,x_{2n-1})dx_{2n}$ where $g$ is a function independent of $x_{2n}$.

We claim that $\gamma^{\prime}$ on $\{x_{2n}=0\}$ has the following properties:

• •

$\gamma^{\prime}(O)\neq 0$;

• •

$d\gamma^{\prime}$ has the maximal possible rank $2n-2$ at $O$;

• •

$\gamma^{\prime}\wedge(d\gamma^{\prime})^{n-1}(O)=0.$

In order to verify these properties, it is sufficient to calculate the constant part $\gamma^{\prime(0)}$ and the linear part $\gamma^{\prime(1)}$ of $\gamma^{\prime}$ at $O$.

Notice that the linear part of the diffeomorphism $\Phi$ in the Straightening Out Theorem for vector fields is just $x_{i}\mapsto x_{i}-\dfrac{c_{i}}{c_{2n}}x_{2n}$ for $i=1,\ldots,2n-1$ and $x_{2n}\mapsto\dfrac{1}{c_{2n}}x_{2n}$. Denote the quadratic part of the old coordinate $x_{n+1}$ in new coordinate system (i.e., the quadratic part of the $(n+1)$-th component of $\Phi^{-1}$) by $Q$. Then the constant part $\gamma^{(0)}$ and the linear part $\gamma^{(1)}$ of $\gamma$ are, respectively,

[TABLE]

Write $Q$ as the sum of a function $Q^{\prime}(x_{1},\ldots,x_{2n-1})$ independent of $x_{2n}$ and a function $Q^{\prime\prime}$ divisible by $x_{2n}$. Remembering that $\gamma^{\prime}$ is also independent of $x_{2n}$, we conclude that the constant part and the linear part of $\gamma^{\prime}$ are, respectively,

[TABLE]

Using these expressions, it is easy to see that the three properties stated above hold.

It follows directly from [20] that $\gamma^{\prime}$ has a normal form $(x_{1}+1)d_{x_{n+1}}+\sum_{i=2}^{n-1}x_{i}dx_{n+i}$ by changing only the coordinates $x_{1},\ldots,x_{2n-1}$. Thus, $\gamma=(x_{1}+1)d_{x_{n+1}}+\sum_{i=2}^{n-1}x_{i}dx_{n+i}+g(x_{1},\ldots,x_{2n-1})dx_{2n}$ and $\dfrac{\partial g}{\partial x_{n}}(O)\neq 0$ since $(d\gamma)^{n}(O)\neq 0$. Then the transformation that maps $x_{n}$ to $g$ and preserves the other coordinates is indeed a local diffeomorphism not affecting the expression $X=\dfrac{\partial}{\partial x_{2n}}$, since $g$ is independent of $x_{2n}$, and keeps $\gamma$ in normal form.

Note that although the position of the coordinate $x_{i}$ in $\gamma$ is not exactly the same for $i=1$, $2\leqslant i\leqslant n$, and $n+1\leqslant i\leqslant 2n$, the argument above works with very few changes. Indeed, if we assume at the beginning $c_{k}\neq 0$ ($k\neq n+1)$ so that $X=\dfrac{\partial}{\partial x_{k}}$, we obtain a similar splitting $\gamma=\gamma^{\prime}+\gamma^{\prime\prime}$ with $\gamma^{\prime}$ having the three properties stated above. (Now the $dx_{k}$ component $\gamma^{\prime\prime}$ is independent of $x_{k}$ and $\gamma^{\prime(1)}=\displaystyle\sum_{i=1,i\neq k}^{n}x_{i}dx_{n+i}+dQ^{\prime}$.) We only need to rename the coordinates after the final normalization of $\gamma$.

If $c_{n+1}\neq 0$, we need to use a supplementary linear transformation before normalizing $\gamma^{\prime}$. The first step is again the straightening out of $X$ to $c_{n+1}\dfrac{\partial}{\partial x_{n+1}}$ by a diffeomorphism $\Phi$ whose linear part is $x_{i}\mapsto x_{i}-\dfrac{c_{i}}{c_{n+1}}x_{n+1}$ for $i=1,\ldots,n,n+2,\ldots,2n$ and $x_{n+1}\mapsto x_{n+1}$. We write again $\gamma=\gamma^{\prime}+\gamma^{\prime\prime}$, where $\gamma^{\prime}$ is a 1-form on $\{x_{n+1}=0\}$ and $\gamma^{\prime\prime}$ has the expression $g(x_{1},\ldots,x_{n},x_{n+2},\ldots,x_{2n})dx_{n+1}$ with $g(0)=1$. Since $\gamma^{\prime}(O)=0$ in this case, we use a linear transformation which maps $x_{n+1}$ to $x_{n+1}-x_{2n}$ and does not affect the other $2n-1$ coordinates. Then $X$ has still the form $c_{n+1}\dfrac{\partial}{\partial x_{n+1}}$ after this linear transformation, while the transformed $\gamma^{\prime}$ no longer vanishes at $O$ (indeed, the constant part of $\gamma^{\prime}$ is $dx_{2n}$). Moreover, we have $(d\gamma^{\prime})^{n-1}(O)\neq 0$ since $(d\gamma)^{n}(O)\neq 0$, where

[TABLE]

We also verify that $\gamma^{\prime}\wedge(d\gamma^{\prime})^{n-1}(O)=0$, which implies, by [20], that $\gamma^{\prime}$ has the normal form $\gamma^{\prime}=(x_{2}+1)dx_{n+2}+\sum_{i=3}^{n}x_{i}dx_{n+i}$ by changing only the coordinates $x_{1},\ldots,x_{n},x_{n+2},\ldots,x_{2n}$. Indeed, the linear part of $\gamma$ is

[TABLE]

where $Q$ is some quadratic function. The linear part $\gamma^{\prime(1)}$ of $\gamma^{\prime}$ can be split into four parts: $\sum_{i=2}^{n-1}x_{i}dx_{n+i}$, the $dx_{2n}$ component, a $1$-form divisible by $x_{2n}$, and a differential form $dQ^{\prime}$ of some quadratic function $Q^{\prime}$. Then $(d\gamma^{\prime(1)})^{n-1}$ is a wedge product of $dx_{2n}$ and some $(2n-3)$-form; therefore, its wedge product with the constant part $\gamma^{\prime(0)}=dx_{2n}$ vanishes.

Finally, we conclude that $\dfrac{\partial g}{\partial x_{1}}(O)\neq 0$, and therefore $x_{1}=g-g(O),x_{2},\ldots,x_{2n}$ forms a coordinate system in which $X=c_{n+1}\dfrac{\partial}{\partial x_{n+1}}$ and $\gamma=(x_{1}+1)dx_{n+1}+\gamma^{\prime}$. Now use a linear transformation to arrive at the desired normalization. ∎

5.1.2. The case $X(O)=0$

Theorem 5.4**.**

Let $\alpha$ be a non-vanishing singular contact form and $X$ a vector field vanishing at $O$ which preserves $\alpha$. Then $X$ and $\alpha$ can be normalized simultaneously, i.e., there is a coordinate system in which $\alpha$ is in the normal form (5.1) and $X$ commutes with its semisimple linear part.

Proof.

We assume that $\alpha$ has been brought into the normal form (5.1). Since $\mathcal{L}_{X}\alpha=0$, regarding $X$ as a vector field on $\{\theta=0\}$ by Lemma 5.2, we interpret $X$ as a Hamiltonian vector field with respect to the standard symplectic form $\omega_{0}=d\alpha$. Thus we have a Hamiltonian function $H$, with $H(O)=0$, i.e, $X\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega_{0}=-dH$. Write

[TABLE]

Since $X(O)=0$, $H(O)=0$, and

[TABLE]

we get $X\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\alpha-H=0$, which, in the given coordinates, takes the form

[TABLE]

By the Taylor expansion, this equation provides two properties of $H$:

1. (1)

$H$ is independent of $x_{1}$, which implies that $x_{n+1}$ is a first integral of $X$; 2. (2)

$H$ is the sum of monomial terms of the form (up to constant coefficients) $\prod_{i=2}^{2n}x_{i}^{\ell_{i}}$ such that $\sum_{i=2}^{n}\ell_{i}=1$.

In particular, the quadratic part $H^{(2)}$ of $H$ is the sum of $c_{ij}x_{i}x_{j}$ in which $c_{ij}$ is constant, $i\in\{2,\ldots,n\}$ and $j\in\{n+1,\ldots,2n\}$. Then the coefficient matrix $A$ of the linear part of $X$ is of the form $\begin{pmatrix}A_{1}&0\\ 0&A_{2}\end{pmatrix}$. Moreover, we can keep the standard form of $\omega_{0}$ and put $A_{1}$ into Jordan normal form by two linear transformations on the complementary $n$-dimensional coordinate planes $\{x_{1},\ldots,x_{n}\}$ and $\{x_{n+1},\ldots,x_{2n}\}$.

Now we claim that the (formal) diffeomorphism $Id_{\theta}\oplus\Phi$ puts $X$ into Poincaré-Dulac normal form without changing $\alpha$, where $Id_{\theta}$ is the identity map for the coordinate $\theta$ and $\Phi$ is the classical symplectic (with respect to $\omega_{0}$) diffeomorphism (constructed step by step below) on $\{\theta=0\}$ normalizing $H$ into the Poincaré-Birkhoff normal form. When $H$ is in the Poincaré-Birkhoff normal form, $X$ is also automatically in the Poincaré-Dulac normal form.

We only need to prove that $\Phi$ preserves the $1$-form $\beta:=\alpha-\theta d\theta$ on $\{\theta=0\}$. Let us verify this assertion by showing that $\Phi$ preserves the associated vector field $Z=(1+x_{1})\dfrac{\partial}{\partial x_{1}}+\sum_{i=2}^{n}x_{i}\dfrac{\partial}{\partial x_{i}}$ of $\beta$ defined by $Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega_{0}=\beta$. Indeed, decomposing the quadratic part $H^{(2)}$ of $H$ into the sum of the semisimple part $H^{s}$ and the nilpotent part $H^{n}$, we see that every resonant term of $H$ lies in the subspace of the kernel of $X_{H^{s}}$. At step $k\geq 3$, we assume that all the non-resonant terms of $H$ of degree smaller than $k$ have been eliminated. In order to eliminate the non-resonant terms of degree $k$, one finds a homogeneous function $F^{(k)}$ of degree $k$ such that the image of $F^{(k)}$ under $X_{H^{(2)}}$ coincides with these non-resonant terms in $H^{(k)}$, i.e., $H^{(k)}-X_{H^{(2)}}F^{(k)}$ contains only resonant terms. Then the time-$1$ flow of $X_{F^{(k)}}$ is a symplectic diffeomorphism $\Phi^{(k)}$ which changes the $k$-th order terms of $H$ into normal form without changing terms of lower degrees. Constructing inductively, from $k=3$ to infinity, we get a formal symplectic diffeomorphism $\Phi$ (equal to the formal infinite composition of the above $\Phi^{(k)}$), which puts $H$ into Poincaré-Birkhoff normal form.

Thanks to the properties of $H$, and the above property of $H^{(2)}$ in particular, $F^{(k)}$ can be chosen such that every monomial term $\prod_{i=1}^{2n}x_{i}^{\ell_{i}}$ also satisfies $\ell_{1}=0$ and $\sum_{i=2}^{n}\ell_{i}=1$. Moreover, this construction guarantees that the higher order terms of $H$ in the new coordinate system after each transformation $\Phi^{(k)}$ have the same properties. Then $X_{F^{(k)}}$ acquires the form

[TABLE]

Since $X_{F^{(k)}}$ commutes with the Euler vector field $E=\sum_{i=1}^{n}x_{i}\dfrac{\partial}{\partial x_{i}}$ and commutes with $\dfrac{\partial}{\partial x_{1}}$, we conclude that it commutes with the associated vector field $Z$ of $\beta$ and, furthermore, it preserves $\beta$. Consequently, $\Phi$ also preserves $\beta$. ∎

5.2. Two types of non-degenerate singular contact forms

We now consider singular contact forms which vanish at $O$, with a generic non-degeneracy condition.

If $O$ is a non-degenerate singularity of the singular contact form $\alpha$ then it is isolated, by definition. Moreover, $O$ must be an equilibrium point of the vector field which preserves the singular contact structure.

Let us present normal forms of singular contact forms in the transversal case and the generic tangential case obtained in [19].

Theorem 5.5**.**

Let $O$ be a non-degenerate singularity of a smooth (resp., real or complex analytic) singular contact 1-form $\alpha$ on a manifold of dimension $2n+1$. Then there is a local smooth (resp., analytic) coordinate system $(\theta,x_{1},\ldots,x_{2n})$ in which $\alpha$ has the expression

[TABLE]

in the transversal case, or the expression

[TABLE]

in the tangential case with a generic tangency, where $\gamma=\sum_{i=1}^{2n}g_{i}dx_{i}$ is a 1-form which is basic with respect to $\dfrac{\partial}{\partial\theta}$, i.e., the functions $g_{i}$ do not depend on $\theta$, and such that

[TABLE]

is a symplectic form in $2n$ variables (which can be put into Darboux canonical form).

The primitive form $\gamma$ can be (formally, or analytically under some Diophantine conditions, or smoothly under hyperbolicity conditions) normalized (over $\mathbb{C}$) in the subspace $(x_{1},\ldots,x_{2n})$ as follows: there are positive integers $n_{1}=1<n_{2}<\cdots<n_{k}<n_{k+1}=n+1$, and eigenvalues $\lambda_{1},\ldots,\lambda_{2n}$ of $X$, such that $\lambda_{i}+\lambda_{n+i}=1$ for all $i=1,\ldots,n$, $\lambda_{i}=\lambda_{j}$ for any $n_{s}\leq i<j<n_{s+1}$ ($s=1,\ldots,k)$, and

[TABLE]

where:

i) $\gamma_{i}$ has the form

[TABLE]

ii) The functions $Q_{i}$ are as follows:

• •

If $\lambda_{i}\neq\frac{1}{2}$ then $Q_{i}=0$ or $Q_{i}=\sum_{j=n_{i}}^{n_{i+1}-2}x_{j+1}x_{n+j}$;

• •

If $\lambda_{i}=\frac{1}{2}$ then $Q_{i}$ belongs to one of the following four cases:

• –

$Q_{i}=0$;

• –

$Q_{i}=x_{n_{i}+n}^{2}$* with $n_{i+1}=n_{i}+1$ in this case;*

• –

$Q_{i}=2\sum_{j=n_{i}}^{n_{i+1}-2}x_{j}x_{n+j+1}+(-1)^{n_{i+1}-n_{i}}x_{n_{i+1}-1}^{2}$;

• –

$Q_{i}=2\sum_{j=n_{i}}^{n_{i+1}-2}x_{j}x_{n+j+1}$* with $n_{i+1}-n_{i}\geq 3$ and is an odd number.*

iii) $R$ is a function of $x_{1},\ldots,x_{2n}$ whose infinite jet at [math] contains only the monomial terms $\prod x_{i}^{\alpha_{i}}$ satisfying the resonance relation

[TABLE]

Note that we reduce the normalization of $\gamma$ to the normalization of its associated vector field $Z$ defined by $Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega_{0}=\gamma$ in [19]. That is, supposing that $\gamma$ is already in normal form in the coordinate system $(\theta,x_{1},\ldots,x_{2n})$, then its associated vector field $Z$ defined by $Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega_{0}=\gamma$ is in Poincaré-Dulac normal form, where $\omega_{0}=d\gamma$ is the canonical symplectic form on $\{\theta=0\}$.

We would like to mention a special example as a direct application of this reduction, which shows the existence of the simultaneous normalization of the dynamical system and the singular contact distribution (not only the form itself).

Example 5.6*.*

The vector field $Z_{1}=\dfrac{1}{2}\theta\dfrac{\partial}{\partial\theta}+Z$ and its semisimple part $Z_{1}^{s}=\dfrac{1}{2}\theta\dfrac{\partial}{\partial\theta}+Z^{s}$ preserve the transversal singular structure. In fact, $Z$ is invariant under its flow $\phi_{Z}^{t}(x)$, and we have $(\phi_{Z}^{t})^{*}\omega=e^{t}\omega$ since $\mathcal{L}_{Z}\omega=\omega$. Therefore, we have $(\phi_{Z}^{t})^{*}\gamma=Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!e^{t}\omega=e^{t}\gamma$. As the flow $\phi_{Z_{1}}^{t}(\theta,x)=(\theta e^{\frac{1}{2}t},\phi_{Z}^{t}(x))$ and the transversal singular contact form $\alpha=\theta d\theta+\gamma$, we have

[TABLE]

Similar computations are also true for the semisimple vector field $Z_{1}^{s}$.

5.2.1. The transversal case

Recall that a singular contact form $\alpha$ in dimension $2n+1$ is called transversal if the kernel of $d\alpha$ is transversal to the hypersurface $S=\{x\ |\ \alpha\wedge(d\alpha)^{n}(x)=0\}$ of singular points of $\alpha$. This transversality condition is generic. In this subsubsection, we study local dynamical systems which preserve this type of singular contact forms.

Theorem 5.7**.**

Suppose a transversal singular contact form $\alpha$ is preserved by a vector field $X$ which vanishes at $O$. Then $\alpha$ and $X$ can be normalized simultaneously, i.e., $\alpha$ is in the form (5.2) as in Theorem 5.5, and $X$ is in Poincaré-Dulac normal form having $\theta$ as a first integral. Moreover, the quadratic function $Q=\sum Q_{i}$ and the higher order function $R$ in (5.2) are first integrals of the semisimple part of $X$.

Proof.

Write $X=f_{0}\dfrac{\partial}{\partial\theta}+\sum_{i=1}^{2n}f_{i}\dfrac{\partial}{\partial x_{i}}$. Then Lemma 5.2 also holds for non-degenerate singular contact forms with identical proof. Thus $\theta$ is a first integral of $X$, $\mathcal{L}_{X}\gamma=0$, and $\omega=d\gamma$ is a symplectic form on $\{\theta=0\}$ (see Theorem 5.5).

Since $X$ is independent of $\theta$, it can be projected to the hyperplane $\{\theta=0\}$. We will show that $X$ commutes with $Z$ defined by $Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega=\gamma$. Indeed, we have

[TABLE]

which implies that $[Z,X]=0.$ Therefore, there exists a new coordinate system in which $Z$ and $X$ are both in Poincaré-Dulac normal forms. Then we use Moser’s path method to normalize $\omega$ to its constant part $\omega^{(0)}$. Moreover, we can choose a path such that this normalization preserves the semisimple parts of $Z$ and $X$. In other words, $Z$ and $X$ remain in Poincaré-Dulac normal form.

Indeed, take $\zeta={Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!(\omega-\omega^{(0)})$, where $Z^{s}$ is the semi-simple part of $Z$, and $\omega_{t}=\omega^{(0)}+t(\omega-\omega^{(0)})$. Then the path of diffeomorphisms, from time [math] to time $1$, given by the flow of the time dependent vector field $Y_{t}$, defined by $Y_{t}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega_{t}=-\zeta$, satisfies our requirements, because

[TABLE]

The third equality above is obtained in the following way. Since $\omega=d\gamma=d(Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega)=\mathcal{L}_{Z}\omega$, it follows that $\mathcal{L}_{Z^{s}}\omega=\omega$ by Theorem 2.3.

It is shown in [19, Lemma 2.2] that $Y_{t}$ commutes with $Z^{s}$. We now show that $Y_{t}$ also commutes with $X^{s}$. Since $X$ preserves $\omega$, we have $\mathcal{L}_{X^{s}}\omega=0$ by the fundamental conservation property (see Theorem 2.3). Clearly, we also have $\mathcal{L}_{X^{s}}\omega^{(0)}=0$ and $\mathcal{L}_{X^{s}}\omega_{t}=0$. Then the following two equations show that $Y_{t}$ also commutes with $X^{s}$:

[TABLE]

Now we use a linear transformation to map $\omega^{(0)}$ into the canonical form $\omega_{0}$ and put, simultaneously, $Z^{s}$ in diagonal form, as in [19]. Then $\gamma$ is also in normal form. The linear transformation does not destroy the Poincaré-Dulac normal form of $X$. (Another way to look at it is to take the associated torus action $\rho$ of the family of commuting vector fields $X$ and $Z$, which contains both the torus action associated to $X$ and the torus action associated to $Z$. A linearization of $\rho$ is a simultaneous normalization of $X$ and $Z$. Then take an equivariant normalization of $\omega$ with respect to $\rho$.)

The rest of the proof is an application of Theorem 2.3; here we give a direct proof.

We can now assume that everything is in normal form and, in particular, $\omega=\omega_{0}$ is in canonical form. Since $X$ commutes with the semisimple part $Z^{s}$ of $Z$, we have

[TABLE]

that is, ${Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\gamma$ is a first integral of $X$. It is also a first integral of the semisimple linear vector field $X^{s}$ since $X$ is in Poincaré-Dulac normal form.

Recall that $Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\gamma=Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!(Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega)=0$, which implies that $\mathcal{L}_{Z}\gamma=Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!d\gamma=Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega=\gamma$, which, in turn, implies that $\mathcal{L}_{Z^{s}}\gamma=\gamma$. Recall also that $\gamma$ is the sum of its linear part $\gamma^{(1)}$ and an exact differential form $dR$, where the $2$-jet of the function $R$ vanishes at $O$. From $\mathcal{L}_{Z^{s}}\gamma=\gamma$ we get $Z^{s}(R)=R$.

Since ${Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\gamma={Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\gamma^{(1)}+Z^{s}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!R={Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\gamma^{(1)}+R$ is a first integral of $X^{s}$, it follows that both its quadratic part ${Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\gamma^{(1)}$ and its higher order part $R$ are first integrals of $X^{s}$. If we abuse the notation and define $\gamma^{s}:={Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega_{0}$, then $\gamma^{(1)}=\gamma^{s}+dQ$. Since ${Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\gamma^{s}={Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!({Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega_{0})=0$, it follows that $\mathcal{L}_{Z^{s}}\gamma^{s}=\gamma^{s}$, hence $\mathcal{L}_{Z^{s}}dQ=dQ$. We conclude that ${Z^{s}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!dQ=\mathcal{L}_{Z^{s}}Q=Q$ is also a first integral of $X^{s}$. ∎

5.2.2. The tangential case

We now assume that $\alpha$ is a tangential non-degenerate singular contact form, i.e., the kernel of $d\alpha$ is tangent to the hypersurface $S=\{x\ |\ \alpha\wedge(d\alpha)^{n}(x)=0\}$ of singular points. We recall that the normal form of $\alpha$ in the generic tangential case is $\alpha=d(\theta^{3}-\theta h(x))+\gamma$ with a non-degenerate function $h$ and normalized primitive $1$-form $\gamma$ (see Theorem 5.5). We have a similar theorem as in the transversal case.

Theorem 5.8**.**

Let $\alpha=d(\theta^{m}-x_{1}\theta)+\gamma$ be a tangential singular contact form preserved by a vector field $X$ having $O$ as its equilibrium point, where $\gamma$ is a primitive $1$-form independent of $\theta$ and $m>2$ is an integer. Then $\alpha$ and $X$ can be normalized simultaneously, i.e., $\alpha=d(\theta^{m}-f(x)\theta)+\gamma$ with $\gamma$ in the normal form given in Theorem 5.5 and $X$ in Poincaré-Dulac normal form having $\theta$ and $f(x)$ as its first integrals. Moreover, the quadratic function $Q=\sum Q_{i}$ and higher order function $R$ in (5.2) are first integrals of the semisimple part of $X$.

In particular, if $\alpha=d(\theta^{3}-x_{1}\theta)+\gamma$ is generically tangent, i.e., the kernel of $d\alpha$ has second order tangency with the singular hypersurface, then $\alpha$ and $X$ can be normalized simultaneously.

Proof.

Assume that $X=f_{0}\dfrac{\partial}{\partial\theta}+\sum_{i=1}^{2n}f_{i}\dfrac{\partial}{\partial x_{i}}$ preserves $\alpha=d(\theta^{m}-x_{1}\theta)+\gamma$. Similar to Lemma 5.2 in the transversal case, there is an analogous lemma (see Lemma 5.9 below) in the tangent singular case, which states that $f_{0}$ and $f_{1}$ are identically zero. Therefore, we have $\mathcal{L}_{X}{d(\theta^{m}-x_{1}\theta)}=\mathcal{L}_{X}\gamma=0$ and the functions $\theta$ and $x_{1}$ are first integrals of $X$.

Using this lemma, the rest of the proof is similar to the one in the transversal case:

• •

$X$, viewed as a vector field on $\{\theta=0\}$, commutes with the associated vector field $Z$ defined by $Z\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!d\gamma=\gamma$; thus $X$ and $Z$ can be put in Poincaré-Dulac normal forms simultaneously;

• •

use Moser’s path method, and then a linear transformation, to normalize $d\gamma$ to the canonical symplectic form without destroying the normal forms of $X$ and $Z$.

Then the primitive $1$-form is automatically in normal form as in Theorem 5.5. Notice that we do not change the variable $\theta$, so the old $x_{1}$ becomes a function $f(x)$ which is still a first integral of $X$ and $df(O)\neq 0$. ∎

Lemma 5.9**.**

Assume that $X=f_{0}\dfrac{\partial}{\partial\theta}+\sum_{i=1}^{2n}f_{i}\dfrac{\partial}{\partial x_{i}}$ preserves $\alpha=d(\theta^{m}-x_{1}\theta)+\gamma$, $m>2$. Then $f_{0}=f_{1}=0$.

Proof.

Since $X$ preserves $\alpha$, it preserves the kernel $\mathbb{K}\dfrac{\partial}{\partial\theta}$ of $d\alpha=\omega$, which means that $f_{1},\ldots,f_{2n}$ are independent of $\theta$.

Consider the component $d\theta$ in $\mathcal{L}_{X}\alpha$. Since $X\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!d\alpha=\sum_{i=1}^{2n}f_{i}\dfrac{\partial}{\partial x_{i}}\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\omega$ does not contain $d\theta$, the $d\theta$-component must come from $dX\!\mathbin{\hbox to6.0pt{\vrule height=0.4pt,width=5.0pt,depth=0.0pt\kern-0.4pt\vrule height=6.0pt,width=0.4pt,depth=0.0pt\hss}}\!\alpha$, more precisely, from the part

[TABLE]

Thus, we have

[TABLE]

Now we can write

[TABLE]

where $C(x)$ is a function independent of $\theta$. The coefficient of $d\theta$ in (5.4) vanishes:

[TABLE]

Let $\ell$ be any non-negative integer and assume $f_{0}$ is divisible by $x_{1}^{\ell}$. Then:

• •

set $\theta=0$ in (5.5) to conclude that $C(x)$ is divisible by $x_{1}^{\ell+1}$;

• •

set $\theta=0$ in (5.6) to conclude that $f_{1}(x)$ is divisible by $x_{1}^{\ell+1}$;

• •

go back to equation (5.5) to conclude that $f_{0}$ is also divisible by $x_{1}^{\ell+1}$.

We proved inductively that $f_{0}$ is divisible by any power of $x_{1}$. Thus $f_{0}$ vanishes if it is a formal or analytic function. So do $f_{1}(x)$ and $C(x)$. ∎

5.2.3. Singular contact forms with integrable systems

We have seen that singular contact forms are not finitely determined and we cannot find an a priori diffeomorphism that brings them in normal forms having finite order jets. However, the complexity of singular contact forms diminishes if they are preserved by some vector fields. The more such vector fields, the less freedom for singular contact forms. So if a singular contact form is preserved by an integrable system, then the singular contact form itself becomes simpler.

Proposition 5.10**.**

Let $\gamma$ be a primitive $1$-form on $(\mathbb{K}^{2n},O)$. Suppose $\gamma$ is preserved by $p$ pairwise commuting vector fields $X_{1},\ldots,X_{p}$ whose semisimple parts of the linear parts are independent almost everywhere. Then $p\leqslant n$. Moreover, if $p=n$, then $\gamma=\gamma^{s}=\sum_{i=1}^{n}(\lambda_{i}x_{i}dx_{n+i}+(\lambda_{i}-1)x_{n+i}dx_{i})$ is linear and the vector fields in the Poincaré-Dulac normal forms have diagonal linear parts.

Proof.

From the proof of Theorem 5.7, we can assume that the primitive $1$-form and the vector fields are all in normal forms, i.e., $\gamma$ is as in Theorem 5.5 and $[X_{i},X_{j}^{s}]=0$ for all $i,j$. By the Conservation Theorem 2.3, $\gamma$ is also preserved by the semisimple parts $X_{1}^{s},\ldots,X_{p}^{s}$ of the vector fields. After a linear transformation, we can assume $X_{i}^{s}=\sum_{k=1}^{n}\mu_{ik}x_{k}\dfrac{\partial}{\partial x_{k}}$ for $i=1,\ldots,p$.

Using again the equation $\mathcal{L}_{X_{i}^{s}}\omega^{(0)}=\mathcal{L}_{X_{i}^{s}}d\gamma=d\mathcal{L}_{X_{i}^{s}}\gamma=0$, where

[TABLE]

is the constant part of the symplectic form $d\gamma$, we get

[TABLE]

Since $X_{1}^{s},\ldots,X_{p}^{s}$ are independent almost everywhere, the rank of the $p\times(2n)$ matrix $(\mu_{ik})$ is $p$ and because $\omega^{(0)}$ is non-degenerate, among all the $c_{jk}$’s there are at least $n$ that do not vanish. Thus (5.7) implies that $p\leqslant n$.

If the number of vector fields is maximal, i.e., $p=n$, then we have exactly $n$ among all of the $c_{jk}$’s that do not vanish. The set of subscripts $j,k$ for the $n$ nonzero $c_{jk}$’s must be included in the set $\{1,2,\ldots,2n\}$ by non-degeneracy. Without loss of generality, we assume that $c_{j\,n+j}\neq 0$ for all $j=1,\ldots,n$. By (5.7), this implies $\mu_{ij}+\mu_{i\,n+j}=0$ for all $i=1,\ldots,p=n$.

The common (formal or analytic) first integrals are generated by the quadratic functions $x_{j}x_{n+j}$. Then any monomial term of $H$ has the expression $\prod_{j=1}^{n}x_{j}^{\ell_{j}}x_{n+j}^{\ell_{j}}$ since $H$ is automatically a first integral. By the resonance relation (5.3) on the indices, we have

[TABLE]

It thus follows that there is only one $\ell_{j}=1$ and the others are zero, which implies that $R=0$ since we request, a priori, that the summands of $R$ have degree at least $3$. Hence, the primitive $1$-form now reads $\alpha=\gamma^{s}+dQ$. Moreover, the quadratic polynomial is also a common first integral of $X_{i}^{s}$’s. Thus $Q=\sum_{j=1}^{n}c_{j}x_{j}x_{n+j}$, where the $c_{j}$’s are constant coefficients. This implies $c_{j}=0$ since $Q$ does not contain any term of the form $x_{j}x_{n+j}$ in our normal form. ∎

We can get similar results for a transversal singular contact form $\alpha$ under the same assumptions. By Theorem 5.7, we can assume that in a suitable coordinate system $(\theta,x_{1},\ldots,x_{2n})$, $\alpha$ is in normal form and the vector fields $X_{1},\ldots,X_{p}$ are all in Poincaré-Dulac normal forms. By an argument similar to that in Lemma 5.2, we can conclude that $X_{1},\ldots,X_{p}$ are independent of $\theta$ and have $\theta$ as a common first integral. Therefore, $X_{1},\ldots,X_{p}$ also preserve the primitive $1$-form $\gamma$ and we have $p\leqslant n$; if $p=n$, then $\gamma$ is linear. This proves the following result.

Proposition 5.11**.**

Let $\alpha$ be a transversal singular contact form on $(\mathbb{K}^{2n+1},O)$. Suppose $\alpha$ is preserved by $p$ pairwise commuting vector fields $X_{1},\ldots,X_{p}$ whose semisimple parts of the linear parts are independent almost everywhere. Then $p\leqslant n$. If $p=n$, then $\alpha=\theta d\theta+\gamma^{s}$ is linear.

We note that the proposition is not true in the generic tangential case. The maximum possible number of such non-degenerate vector fields is $n-1$ since we can exclude the situation $p=n$. Otherwise, we use Lemma 5.9 and then the vector fields preserve the primitive $1$-form $\gamma$, which guarantees that the common first integrals of the semisimple parts of the vector fields are functions of the $x_{i}x_{n+i}$’s. On the other hand, we request $f(x)$ in $\alpha=d(\theta^{3}-f\theta)+\gamma$ to be a first integral of the semisimple parts of the vector fields, which contradicts the non-degeneracy of $f(x)$, in the sense that $df(0)\neq 0$.

Acknowledgements

Part of this work was done during Nguyen Tien Zung’s visits to Shanghai Jiaotong University (SJTU) in 2018 and 2019 under a “High-End Foreign Experts Project” of China. He thanks the members of the SJTU School of Mathematical Sciences, especially Prof. Xiang Zhang and Jie Hu, for their warm hospitality and excellent working conditions.