St\"ackel transform of Lax equations

TL;DR

This paper develops a method to generate Lax pairs for a broad class of St"ackel systems using the multi-parameter St"ackel transform, providing a parametrized family of solutions.

Contribution

It introduces a systematic way to construct Lax pairs for St"ackel systems via the St"ackel transform, expanding the tools for integrability analysis.

Findings

Constructed Lax pairs for various St"ackel systems.

Parametrized Lax pairs by an arbitrary function.

Enhanced understanding of integrability structures.

Abstract

We construct Lax pairs for a wide class of St\"ackel systems by applying the multi-parameter St\"ackel transform to Lax pairs of a suitably chosen systems from the seed class. For a given St\"ackel system, the obtained set of non-equivalent Lax pairs is parametrized by an arbitrary function.

Equations

$$\sigma (\lambda )+\sum _{j=1}^n{\tilde{H}}_j{\lambda }^{{\gamma }_j}=\frac{1}{2}f(\lambda ){\mu }^2\text{, }$$

$$\sigma(\lambda)+\sum_{j=1}^{n}H_{j}\lambda^{n-j}=\frac{1}{2}f(\lambda)\mu^{2},\ \$$

$$h_i=h_i(\xi ,{\alpha }_1,\dots ,{\alpha }_k),i=1,\dots ,n$$

$$h_{s_i}(\xi ,{\alpha }_1,\dots ,{\alpha }_k)={\tilde{\alpha }}_i,i=1,\dots ,k$$

$${\alpha }_i={\tilde{h}}_{s_i}(\xi ,{\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_k),i=1,\dots ,k,$$

$${\tilde{h}}_i=h_i{|}_{{\alpha }_1\to {\tilde{h}}_{s_1},\dots ,{\alpha }_k\to {\tilde{h}}_{s_k}},i=1,\dots ,n,i\notin \{s_1,\dots ,s_k\}.$$

$$\frac{d\xi }{d{t}_i}=\pi d{h}_i\equiv X_i\text{, }i=1,\dots ,n$$

$$\frac{d\xi }{d{\tilde{t}}_i}=\pi d{\tilde{h}}_i\equiv {\tilde{X}}_i\text{, }i=1,\dots ,n$$

$$M_{\alpha ,\tilde{\alpha }}=\left\{\xi \in M:h_{s_i}(\xi ,{\alpha }_1,\dots ,{\alpha }_k)={\tilde{\alpha }}_i\text{, }i=1,\dots k\right\}$$

$$M_{\alpha ,\tilde{\alpha }}=\left\{\xi \in M:{\tilde{h}}_{s_i}(\xi ,{\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_k)={\alpha }_i\text{, }i=1,\dots k\right\}$$

$$h_{s_i}(\xi ,{\tilde{h}}_{s_1}(\xi ,{\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_n),\dots ,{\tilde{h}}_{s_k}(\xi ,{\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_n))\equiv {\tilde{\alpha }}_i\text{, }i=1,\dots ,k$$

$${\tilde{h}}_i(\xi ,{\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_n)\equiv h_i(\xi ,{\tilde{h}}_{s_1}(\xi ,{\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_n),\dots ,{\tilde{h}}_{s_k}(\xi ,{\tilde{\alpha }}_1,\dots ,{\tilde{\alpha }}_n)),i=1,\dots ,n\text{, }i\notin \{s_1,\dots ,s_k\}$$

$$d{h}_{s_i}=-\sum _{j=1}^k\frac{\partial h_{s_i}}{\partial {\alpha }_j}d{\tilde{h}}_{s_j}\text{, }i=1,\dots ,k$$

$$d{h}_{{}_i}=d{\tilde{h}}_i-\sum _{j=1}^k\frac{\partial h_i}{\partial {\alpha }_j}d{\tilde{h}}_{s_j}\text{, }i=1,\dots ,n,i\notin \{s_1,\dots ,s_k\}\text{ .}$$

$$dh=Ad\tilde{h}$$

$$A_{ij}={\delta }_{ij}\text{ for }j\notin \{s_1,\dots ,s_k\}\text{, }{A}_{i{s}_j}=-\frac{\partial h_i}{\partial {\alpha }_j}\text{ for }j=1,\dots ,k$$

$$\det A=\pm \det \left(\frac{\partial h_{s_i}}{\partial {\alpha }_j}\right)$$

$$X_{s_i}$$

$$X_{{}_i}$$

$$X=A\tilde{X}$$

$$\xi =\xi (t_1,\dots ,t_n,{\xi }_0)$$

$$d\tilde{t}=A^{T}dt$$

$$\tilde{\xi }({\tilde{t}}_1(t_1,\dots ,t_n,{\xi }_0),\dots ,{\tilde{t}}_n(t_1,\dots ,t_n,{\xi }_0),{\xi }_0)=\xi (t_1,\dots ,t_n,{\xi }_0)$$

$$L_{t_j}=\left[U_j,L\right]\text{, }j=1,\dots ,n$$

$${\tilde{L}}_{{\tilde{t}}_j}=[{\tilde{U}}_j,\tilde{L}]\text{, }j=1,\dots ,n$$

$$\begin{array}[c]{c}\tilde{L}(\lambda,\xi,\tilde{\alpha})=L(\lambda,\xi,\tilde{h}(\xi,\tilde{\alpha}))\\ \tilde{U}_{j}(\lambda,\xi,\tilde{\alpha})={\displaystyle\sum\limits_{i=1}^{n}}(A^{-1})_{ji}(\xi,\tilde{h}(\xi,\tilde{\alpha}))U_{i}(\lambda,\xi,\tilde{h}(\xi,\tilde{\alpha}))\text{\ }\end{array}$$

$${\partial }_{{\tilde{t}}_j}{\tilde{h}}_i=0$$

$$\frac{\partial }{\partial {\tilde{t}}_j}=\sum _{i=1}^n(A^{-1}{)}_{ji}\frac{\partial }{\partial t_i},j=1,\dots ,n.$$

$${\tilde{L}}_{{\tilde{t}}_j}(\lambda ,\xi ,\tilde{\alpha })$$

$$=\sum _{i=1}^n{A}^{-1}(\xi ,\alpha {)}_{ji}\left[U_i(\lambda ,\xi ,\alpha ),L(\lambda ,\xi ,\alpha )\right]=\left[{\tilde{U}}_j(\lambda ,\xi ,\tilde{\alpha }),\tilde{L}(\lambda ,\xi ,\tilde{\alpha })\right]$$

$$\sigma (\lambda )+\sum _{j=1}^n{H}_j{\lambda }^{n-j}=\frac{1}{2}f(\lambda ){\mu }^2.$$

Full text

Keywords and phrases: Hamiltonian systems, completely integrable systems, Stäckel systems, Stäckel transform, Lax representation

1 Introduction

In this paper we show how to construct Lax pairs for Stäckel systems generated by spectral curves of the form

[TABLE]

where $\gamma_{i}\in\mathbb{N}$ with the normalization $\gamma_{1}>\gamma_{2}>\cdots>\gamma_{n}=0$ and where $\sigma,$ $f$ are Laurent polynomials in $\lambda$. Taking  $n$ copies of this curve at points $(\lambda_{i},\mu_{i})$ we obtain a system of $n$ linear equations for $\tilde{H}_{j}$ and its solution yield $n$ commuting, in the sense of canonical Poisson bracket $\left\{\lambda_{i},\mu_{j}\right\}=\delta_{ij}$, Stäckel Hamiltonians $\tilde{H}_{j}(\lambda,\mu)$. Such Stäckel systems are fairly general. Each choice of the constants $\gamma=\left\{\gamma_{1},\cdots,\gamma_{n}\right\}$ fixes the Stäckel matrix $S_{ij}=\lambda_{i}^{\gamma_{j}}$ in (1.1). Unless we specify the functions $\sigma$ and $f$ the curve (1.1) yields a family of systems that we will call a $\gamma$-class. Choosing the constants $\gamma_{j}$ as $\gamma_{j}=n-j$, $j=1,\ldots,n$ yields the special $\gamma$-class

[TABLE]

which is called the Benenti class [1, 2, 3] or the seed class.

In paper [6] we demonstrated how to generate the Hamiltonians $\tilde{H}_{i}$ of the $\gamma$-class (1.1) as a multi-parameter Stäckel transform [10, 9, 12, 13, 11, 5, 7] of Hamiltonians $H_{i}$ from the class (1.2). Also, in the recent paper [8] the authors found Lax pairs $(L(\lambda),U_{i}(\lambda))$, $i=1,\ldots,n$ for equations generated by the Hamiltonians given by (1.2). Combining the ideas of these papers we will find Lax pairs $(\tilde{L}(\lambda),\tilde{U}_{i}(\lambda))$, $i=1,\ldots,n$ for all the Hamiltonian systems of a given $\gamma$-class (1.1). In order to do this we will make an appropriate extension of Hamiltonians from both classes (1.1) and (1.2) by a number of parameters $\alpha_{i}$ and $\tilde{\alpha}_{i}$ so that the extended systems will be related by a multi-parameter Stäckel transform. Applying this Stäckel transform to the Lax pairs $(L(\lambda,a),U_{i}(\lambda,a))$ of the parameter-dependent systems from the seed class, we will obtain the Lax pairs $(\tilde{L}(\lambda,\tilde{a}),\tilde{U}_{i}(\lambda,\tilde{a}))$ of the parameter-dependent Hamiltonians of $\gamma$-class. Finally, by setting all the parameters to zero we will obtain the sought Lax pairs $(\tilde{L}(\lambda),\tilde{U}_{i}(\lambda))$ for (1.1).

The paper is organized as follows. In Section 2 we remind the main concepts of the multi-parameter Stäckel transform. In Section 3 we prove Theorem 2 describing how Lax pairs transform under the Stäckel transform. Section 4 is devoted to Hamiltonian systems and their Lax representations $(L(\lambda),U_{i}(\lambda))$ from the seed class (1.2). In Section 5 we remind the Stäckel transform relating systems from the seed class (1.2) and these from arbitrary $\gamma$-class (1.1). Finally, in Section 6, we apply the results of Section 3 and Section 5 in order to construct the Lax pairs for the Stäckel systems from $\gamma$-classes (1.1). The paper is furnished with few examples.

2 Stäckel transform of integrable Hamiltonian systems

Let us consider a Liouville integrable Hamiltonian system on a $2n$-dimensional Poisson manifold $(M,\pi)$ defined by $n$ Hamiltonians $h_{i}:M\rightarrow\mathbb{R}$ on $M,$ each depending on $k\leq n$ parameters $\alpha_{1},\dots,\alpha_{k}$ so that

[TABLE]

where $\xi\in M$. From $n$ functions in (2.1) we choose $k$ functions $h_{s_{i}}$, $i=1,\ldots,k$, where $\{s_{1},\dots,s_{k}\}$ $\subset\{1,\dots,n\}$. Solving (we assume it is globally possible) the system of equations

[TABLE]

(where $\tilde{\alpha}_{i}$ is another set of $k$ parameters) with respect to $\alpha_{i}$ yields

[TABLE]

where the right-hand sides of these solutions define $k$ new functions $\tilde{h}_{s_{i}}$ on $M$, each depending on $k$ parameters $\tilde{\alpha}_{i}$. Let us also define $n-k$ functions $\tilde{h}_{i}$ by substituting $\tilde{h}_{s_{i}}$ instead of $\alpha_{i}$ in $h_{i}$ for all $i\notin\{s_{1},\dots,s_{k}\}$

[TABLE]

The functions $\tilde{h}_{i}=\tilde{h}_{i}(\xi,\tilde{\alpha}_{1},\dots,\tilde{\alpha}_{k}),$ $i=1,\dots,n$, defined through (2.3) and (2.4) are called the $k$-parameter Stäckel transform of the functions (2.1). If we perform again the Stäckel transform on the functions $\tilde{h}_{i}$ with respect to $\tilde{h}_{s_{i}}$ we will receive back the functions $h_{i}$ in (2.1). One can prove [11, 6] that Stäckel transform preserves functional independence as well as involutivity with respect to $\pi$.

The Hamiltonians $h_{i}$ yield $n$ commuting Hamiltonian systems on $M$

[TABLE]

depending on $k$ parameters $\alpha_{i}$, while $\tilde{h}_{i}$ define $n$ commuting systems

[TABLE]

depending on $k$ parameters $\tilde{\alpha}_{i}$.

Observe that as soon as we fix the values of both all $\alpha_{i}$ and all $\tilde{\alpha}_{i}$ the relation (2.2) defines the $(2n-k)$-dimensional submanifold $M_{\alpha,\tilde{\alpha}}$ given by (2.2):

[TABLE]

or equivalently by (2.3)

[TABLE]

Remark 1

Through each point $\xi$ in M\,\there passes infinitely many submanifolds $M_{\alpha,\tilde{\alpha}}$. If we fix the values of all the parameters $\alpha_{i}$ we can for any $\xi$ always find some values of the parameters $\tilde{\alpha}_{i}$ so that $\xi\in M_{\alpha,\tilde{\alpha}}$ and vice versa, if we fix $\tilde{\alpha}_{i}$, for any given $\xi$ we can find $\alpha_{i}$ so that $\xi\in M_{\alpha,\tilde{\alpha}}$.

As it follows from (2.2), (2.3) and (2.4) the following identities are valid on the whole $M$ and for all values of parameters $\tilde{\alpha}_{i}$:

[TABLE]

[TABLE]

Differentiating (2.9) with respect to $\xi$ we find that on each $M_{\alpha,\tilde{\alpha}}$

[TABLE]

while differentiation of (2.10) gives that on $M_{\alpha,\tilde{\alpha}}$ we have

[TABLE]

The transformations (2.11), (2.12) on $M_{\alpha,\tilde{\alpha}}$ can be written in a matrix form as

[TABLE]

where we denote $dh=(dh_{1},\ldots,dh_{n})^{T}$ and $d\tilde{h}=(d\tilde{h}_{1},\ldots,d\tilde{h}_{n})^{T}$ and where the $n\times n$ matrix $A$ is given by

[TABLE]

From the structure of the matrix $A$ it follows that

[TABLE]

so that $\det A\neq 0$ due to our assumptions. Thus, the relation (2.13) can be inverted yielding $d\tilde{h}=A^{-1}dh$. The relation (2.13) and its inverse can be used to show the functional independence of $\tilde{h}_{i}$ for all values of $\tilde{\alpha}_{i}$ from the functional independence of $h_{i}$ for all values of $\alpha_{i}\,$. Moreover, the same relations are used to prove the involutivity of $\tilde{h}_{i}$ from involutivity of $h_{i}$. See [11, 6] for details.

Since $X_{i}=\pi dh_{i}$ and $\tilde{X}_{i}=\pi d\tilde{h}_{i}$ we obtain from (2.11)-(2.12) that the Hamiltonian vector fields $X_{i}=\pi dh_{i}$ and $\tilde{X}_{i}=\pi d\tilde{h}_{i}$ are on the appropriate $M_{\alpha,\tilde{\alpha}}$ related by the following transformation

[TABLE]

This means that the Hamiltonian vector fields $X_{i}$ and $\tilde{X}_{i}$ span on each $M_{\alpha,\tilde{\alpha}}$ the same $n$-dimensional distribution and also that the vector fields $X_{s_{i}}$ and $\tilde{X}_{s_{i}}$ span on each $M_{\alpha,\tilde{\alpha}}$ the same $k$-dimensional subdistribution of the above distribution. The transformation (2.15)-(2.16) on $M_{\alpha,\tilde{\alpha}}$ can be written in matrix form as

[TABLE]

where we denote $X=(X_{1},\ldots,X_{n})^{T}$ and $\tilde{X}=(\tilde{X}_{1},\ldots,\tilde{X}_{n})^{T}$ and where the $n\times n$ matrix $A$ is given above.

All the vector fields $X_{i}$ and $\tilde{X}_{i}$ are naturally tangent to the corresponding $M_{\alpha,\tilde{\alpha}}$ so that if $\xi_{0}\in$ $M_{\alpha,\tilde{\alpha}}$ then the multi-parameter (simultaneous) solution

[TABLE]

of all equations in (2.5) starting at $\xi_{0}$ for $t=0$, will always remain in $M_{\alpha,\tilde{\alpha}}$ and the same is also true for multi-parameter solutions of (2.6).

The relations (2.15)-(2.16) can be reformulated in the dual language, that of reciprocal (multi-time) transformations. The reciprocal transformation $\tilde{t}_{i}=\tilde{t}_{i}(t_{1},\ldots,t_{n},\xi),$ $i=1,\ldots,n$ given on $M_{\alpha,\tilde{\alpha}}$ by

[TABLE]

where $dt=(dt_{1},\ldots,dt_{n})^{T}$ and $d\tilde{t}=(d\tilde{t}_{1},\ldots,d\tilde{t}_{n})^{T}$, transforms the $k$-parameter solutions (2.18) of the system (2.5) to the $k$-parameter solutions $\tilde{\xi}=\tilde{\xi}(\tilde{t}_{1},\ldots,\tilde{t}_{n},\xi_{0})$ of the system (2.6) (with the same initial condition $\xi(0)=\xi_{0}\in M_{\alpha,\tilde{\alpha}}$) in the sense that for any $\xi_{0}\in M_{\alpha,\tilde{\alpha}}$ we have

[TABLE]

for all values of $t_{i}$ sufficiently close to zero.

The transformation (2.19) is well defined since the right-hand side of (2.19) is an exact differential, as it follows from the above construction. It means that it is possible (at least locally) to integrate (2.19) and obtain an explicit transformation $\tilde{t}_{i}=\tilde{t}_{i}(t_{1},\ldots,t_{n},\xi)$ that takes multi-time (simultaneous) solutions of all Hamiltonian systems (2.5) to multi-time solutions of all the systems in (2.6).

3 Stäckel transform of Lax equations

In the theorem below, we establish a connection between the Lax pairs of the systems related by a multi-parameter Stäckel transform.

Theorem 2

Suppose that the Liouville integrable system (2.5) has the Lax representation

[TABLE]

where $L=L(\lambda,\xi,\alpha)$ and $U_{j}=U_{j}(\lambda,\xi,\alpha)$ are some matrices depending on the spectral parameter $\lambda$. Then the Liouville integrable system (2.6) has the Lax representation

[TABLE]

where

[TABLE]

Thus, in order to obtain the Lax matrix $\tilde{L}(\lambda,\xi,\tilde{\alpha})$ of the system (2.6) it is enough to replace each $\alpha_{i}$ in $L(\lambda,\xi,\alpha)$ by the corresponding $\tilde{h}_{s_{i}}(\xi,\tilde{\alpha})$; the same substitutions are performed in $(A^{-1})_{ji}(\xi,\alpha)$ and in $U_{i}(\lambda,\xi,\alpha)$ in the second formula in (3.3) in order to obtain $\tilde{U}_{j}(\lambda,\xi,\tilde{\alpha})$.

Proof. Fix arbitrary values of the parameters $\tilde{\alpha}_{i}$ and choose a point $\xi\in M$. According to Remark 1 we can then find values of the parameters $a_{i}$ so that $\xi\in M_{\alpha,\tilde{\alpha}}$ and then $\alpha_{i}=\tilde{h}_{s_{i}}(\xi,\tilde{\alpha})$ for $i=1,\ldots,k$. Obviously

[TABLE]

for all $i,j$ and moreover, due to (2.17), we have on $M_{\alpha,\tilde{\alpha}}$

[TABLE]

In consequence, at the chosen (and thus arbitrary) $\xi\in M$

[TABLE]

Let us make two comments on the above theorem. Firstly, the Lax pairs (3.1) and (3.2) are understood as differential-algebraic consequences of the systems (2.5) and (2.6) respectively, i.e. we do not require that the Lax pairs (3.1) and (3.2) actually reconstruct the systems themselves (see also Remark 3). Further, this theorem is a global result, not just restricted to some submanifold $M_{\alpha,\tilde{\alpha}}$.

4 Hamiltonian systems from the seed class and their Lax

representations

Let us now consider separable systems generated by separation curves (spectral curves) in the form

[TABLE]

Solving the system of $n$ copies of (4.1), with $\lambda_{i}$ and $\mu_{i}$ substituted for $\lambda$ and $\mu$, $i=1,\ldots,n$, with respect to $H_{j}$ we obtain $n$ separable (and thus Liouville integrable) Hamiltonians $H_{j}(\lambda,\mu)$ and $n$ related vector fields $X_{j}(\lambda,\mu),$

[TABLE]

on the Poisson manifold $(M,\pi),$ where $E_{j}(\lambda,\mu)$ represent geodesic part of the Hamiltonians and $V_{j}^{(\sigma)}(\lambda)$ represent the potential part. Throughout the article $(\lambda,\mu)=(\lambda_{1},\ldots,\lambda_{n},\mu_{1},\ldots,\mu_{n})$ denotes Darboux (canonical) coordinates on $(M,\pi)$ which are also separation coordinates for all $H_{j}$ [3]. The functions $V_{j}^{(\sigma)}$ are linear combinations of so called basic separable potentials $V_{j}^{(\beta)}$, generated by monomials $\sigma(\lambda)=\lambda^{\beta}$, $\beta\in\mathbb{Z}$. The potentials $V_{j}^{(\beta)}$ can be obtained by solving $n$ copies of the equation

[TABLE]

Explicitly the functions $V_{j}^{(\beta)}$ can be calculated from the formula [5]

[TABLE]

where

[TABLE]

$\rho_{i}(\lambda)=(-1)^{i}\sigma_{i}(\lambda)$ and $\sigma_{i}(\lambda)$ are elementary symmetric polynomials. Notice that for $\beta=0,...,n-1$

[TABLE]

It has been proved in [8] that each system (4.2) has a family of Lax representations

[TABLE]

parametrized by arbitrary nonvanishing smooth functions $g(\lambda)$. The Lax matrix in the Darboux variables $(\lambda,\mu)$, has the form [8]

[TABLE]

while the auxiliary matrices $U_{k}(\lambda)$ in (4.7) are of the form

[TABLE]

where

[TABLE]

and

[TABLE]

(notice that $u(\lambda_{i})=0$ and $v(\lambda_{i})=g(\lambda_{i})\mu_{i}$) while

[TABLE]

where $F(x,y)=\frac{1}{2}f(x)y^{2}-\sigma(x)$. The symbol $[\frac{{}}{{}}]_{+}$ denotes a polynomial part (Laurent polynomial part) of the quotient, i.e. if $P(\lambda)$ is a polynomial or a Laurent polynomial and if $Q(\lambda)$ is a polynomial then:

[TABLE]

where $R(\lambda)$ is a reminder of the quotient, so $\deg R<\deg Q$ (see [8] for details). In particular, for positive basic separable potentials $V_{j}^{(n+s)}$, $s\in\mathbb{N}$ we have

[TABLE]

while for basic negative separable potentials $V_{j}^{(-s)}$, $s\in\mathbb{N}$

[TABLE]

Notice also that the function $w(\lambda)$ in (4.12) splits into kinetic part $w_{E}(\lambda)$ and potential part $w_{V}(\lambda)$ respectively:

[TABLE]

Remark 3

The Lax matrix $L(\lambda)$ (4.8) reconstructs the separation curve (4.1) in the sense that [8]

[TABLE]

The Lax matrices for different choices of $g(\lambda)$ are not equivalent.

The Lax pairs (4.7), although given here in the separation coordinates $(\lambda,\mu)$, are invariant with respect to any change of coordinates on the manifold $M$. In particular, we will use so called Vieté coordinates defined as

[TABLE]

In these coordinates [8]

[TABLE]

and

[TABLE]

5 Stäckel transform of the seed class

In the previous sections we have discussed systems from the seed class, generated by separation curves (4.1), and their Lax pairs (4.7). In this section we will demonstrate, using this knowledge, how to generate systems from $\gamma$-classes (1.1), defined by the separation curves of the form

[TABLE]

where $\gamma_{i}\in\mathbb{N}$ and $\gamma_{1}>\gamma_{2}>\cdots>\gamma_{n}=0$. Actually, our goal is to demonstrate how to construct the Stäckel systems of a given $\gamma$-class (5.1) by applying the multi-parameter Stäckel transform (2.3)-(2.4) to an appropriate system from the seed class (4.1).

In order to be able to relate the systems from classes (4.1) and (5.1) by a Stäckel transform, we need to extend both of them to appropriate multi-parameter systems.

Theorem 4

Assume that $\gamma_{1}>\gamma_{2}>\cdots>\gamma_{k}>n-1$ and S=\{s_{1},\dots,s_{k}\}\subset\{1,\dots,n\}\ with $s_{1}<\cdots<s_{k}$. Then, the Stäckel transform (2.3)-(2.4) transforms the Hamiltonians

[TABLE]

defined by the separation curve from the seed class

[TABLE]

to the Hamiltonians

[TABLE]

defined by the separation curve from the $\gamma$-class $\gamma=\left\{\gamma_{1},\ldots,\gamma_{k}\right\}\cup\left\{n-i:i\notin S\right\}$

[TABLE]

Moreover, the explicit transform between Hamiltonians from both classes takes the form

[TABLE]

where

[TABLE]

and  $h=(h_{1},\ldots,h_{n})^{T}$, $H=(H_{1},\ldots,H_{n})^{T}$, $\alpha=(\alpha_{1},...,\alpha_{k},0,...,0)^{T}$ and likewise for $\tilde{h}$, $\tilde{H}$, $\tilde{\alpha}$. The $n\times n$ matrix $A_{\gamma}$ is given by

[TABLE]

and in particular we obtain the explicit map between $H$ and $\tilde{H}$:

[TABLE]

This theorem follows from Theorem 3 in [6]. A careful inspection of this theorem reveals that the enumeration of Hamiltonians $\tilde{h}_{i}$ in this theorem has been changed in comparison with the general construction presented in Section 2. We still have that $h_{s_{i}}=\tilde{\alpha}_{i}$ for $i=1,\ldots,k$ but now $\tilde{h}_{i}=\alpha_{i}$ for $i=1,\ldots,k$ (and not $\tilde{h}_{{}_{s_{i}}}=\alpha_{i}$ as in the general construction) while the remaining transformed Hamiltonians $\tilde{h}_{i}$ (for $i=k+1,\ldots,n$) are obtained by substituting, in the consecutive $h_{j}$ for $j\notin S$, all $\alpha_{i}$, $i=1,\ldots,k$, with the corresponding $\tilde{h}_{i}$, as the general idea of Stäckel transform stipulates. This is done in order to obtain a convenient enumeration of $\tilde{h}_{i}$ in the transformed system (5.5).

Notice that due to (5.2) the matrix $A_{\gamma}$ is simply the matrix $A$ (2.14) written in the particular settings of this theorem. Notice also that (5.8) is valid on the whole $M$, in contrast with the relations (2.13) that are valid only on $M_{\alpha,\tilde{\alpha}}$. However, by explanations in Section 2, the solutions of (5.2) and (5.4) are related only on the appropriate submanifolds $M_{a,\tilde{\alpha}}$. Thus, although the Stäckel transform (5.8) transforms the parameter-free Liouville-integrable system (4.1) into another parameter-free Liouville integrable system (5.1), the solutions of these systems are not globally related by any reciprocal transformation.

Let us demonstrate the whole procedure on two examples, both involving one-parameter Stäckel transform. We restrict ourselves to one-parameter examples as the examples involving two parameters lead to large and complicated expressions not very suitable to be presented in a printed form.

Example 5

As a first example, consider the Hénon-Heiles system given by the separation curve

[TABLE]

(so that $n=2$, $\sigma(\lambda)=-\lambda^{4}$ and $f(\lambda)=\lambda$) and its one-parameter (so that $k=1$) Stäckel transform (”one-hole deformation” [3])) with $\gamma_{1}=2$ and with respect to the first Hamiltonian, i.e. $S=\left\{1\right\}$. Thus $\gamma=\{2,0\}$ and the extended Hénon-Heiles system is generated by separation curve

[TABLE]

The Stäckel transform (5.6) yields the $\gamma$-system generated by

[TABLE]

Setting $\tilde{\alpha}=0$ we obtain the system generated by

[TABLE]

The matrix $A_{\gamma}$ given by (5.7) is

[TABLE]

and thus the parameter-free Stäckel transform (5.8) between both systems is

[TABLE]

We will illustrate the structure of the above objects in the flat orthogonal coordinates $(x,y)$ of the system. They are given by

[TABLE]

with the conjugate momenta given by

[TABLE]

In the flat coordinates the Hamiltonians $H_{i}$ attain the form

[TABLE]

while their Stäckel transform becomes

[TABLE]

Example 6

Consider now the one-parameter (again $k=1$) Stäckel transform of the system defined by separation curve (4.1)

[TABLE]

(so that $n=3$, $\sigma(\lambda)=\lambda^{5}$ and $f(\lambda)=1$), with $\gamma_{1}=3$ and with respect to the second Hamiltonian, i.e. $S=\{2\}$. Thus $\gamma=\left\{3,2,0\right\}$ so the extended system is defined by separation curve (5.3)

[TABLE]

and the Stäckel transform (5.6) yields the $\gamma$-system defined by (5.5)

[TABLE]

Setting $\tilde{\alpha}=0$ we obtain a new system defined by separation curve

[TABLE]

As

[TABLE]

so the parameter-free Stäckel transform (5.8) between both systems attains the form:

[TABLE]

As in the previous example, we will explicitly illustrate the structure of both systems in another coordinates. This time we make a point transformation to Vieté coordinates (4.18). In these coordinates all the Hamiltonians $H_{i}$ are polynomials [4]. Explicitly

[TABLE]

while

[TABLE]

6 Lax representation of $\gamma$-classes

In this section we apply the results from Section 3 and Section 5 in order to construct the Lax pairs for the Stäckel system (5.1) from a given $\gamma$-class. In order to do this we start from the Lax pairs $(L(\lambda,\alpha)$, $U_{j}(\lambda,\alpha))$ for the extended systems from the seed class (5.3), where

[TABLE]

with

[TABLE]

and $U_{j}(\lambda,\alpha)$ are given by (4.9) with $L(\lambda)$ replaced by $L(\lambda,\alpha)$. From Theorem 2 we obtain the following corollary.

Corollary 7

For any smooth nonvanishing function $g(\lambda)$ the matrices $\left(\tilde{L}(\lambda,\tilde{\alpha}),\tilde{U}_{j}(\lambda,\tilde{\alpha})\right)$ given by

[TABLE]

where

[TABLE]

and with

[TABLE]

constitute the Lax pairs for the extended systems from the $\gamma$-class (5.5).

The matrices $U_{i}(\lambda,\tilde{h}(\xi,\tilde{\alpha}))$ can effectively be calculated by the formulas (4.9) with $L(\lambda)$ replaced by $\tilde{L}(\lambda,\tilde{\alpha})$. Notice that the matrix $A$ in the above formula does not depend on $\alpha$ (it still does depend on $\xi$). Finally, the Lax pairs $\left(\tilde{L}(\lambda),\tilde{U}_{j}(\lambda)\right)$ for the system (5.5) are obtained by letting $\tilde{\alpha}_{i}=0$ in $\left(\tilde{L}(\lambda,\tilde{\alpha}),\tilde{U}_{j}(\lambda,\tilde{\alpha})\right)$.

Example 8

(Example 5 continued) The Hénon-Heiles system (5.9) has the Lax pairs given by (4.8-4.16). Then for the extended system (5.10) we get

[TABLE]

(and $U_{j}$ as given by (4.9)) and thus, for the extended $\gamma$-system (5.11) the Lax matrices are

[TABLE]

and

[TABLE]

so that the Lax matrices for the transformed system (5.12) (or (5.16)) are

[TABLE]

and

[TABLE]

As in Example 5, we present the explicit form of these matrices for a particular function $g(\lambda)$ and in the flat coordinates $(x,y)$, given by (5.13)-(5.14). Thus, for the Hénon-Heiles system (5.9) (or equivalently (5.15)), the Lax matrix $L(\lambda)$ for $g(\lambda)=1$ takes the form [8]

[TABLE]

while

[TABLE]

The Stäckel transform (3.3) of these Lax pairs yields

[TABLE]

and

[TABLE]

i.e. the $g=0$ Lax representation for the system (5.12) (or (5.16)). In a similar way, the $g(\lambda)=\lambda$ Lax representation for the Hénon-Heiles system takes the form

[TABLE]

while the Stäckel transform (3.3) gives the $g(\lambda)=\lambda$ Lax representation

[TABLE]

[TABLE]

for the system (5.12) (or (5.16)).

Example 9

(Example 6 continued). The system (5.17) has the Lax pairs given by (4.8) and (4.9) with $f(\lambda)=1$; let us now also choose $g(\lambda)=1$. In the first step we construct the Lax pairs $\left(L(\lambda,\alpha),U_{j}(\lambda,\alpha)\right)$ for the extended system (5.18). We obtain

[TABLE]

while $U_{j}(\lambda,\alpha)$ are then given by (4.9) with $L(\lambda,\alpha)$ given by (6.3). Thus,

[TABLE]

and

[TABLE]

[TABLE]

As in Example 6, we present the explicit form of these formulas in the Vieté coordinates (4.18). For seed system, generated by Hamiltonians (5.21), the $g(\lambda)=1$ Lax operator $L(\lambda)$ becomes

[TABLE]

while

[TABLE]

[TABLE]

The Stäckel transform (3.3) of the above Lax pairs yields

[TABLE]

and

[TABLE]

i.e. the respective Lax pairs for transformed system generated by Hamiltonians (5.22).

7 Acknowledgments

MB wishes to express his gratitude for Department of Science, Linköping, University, Sweden, for their kind hospitality.