Integrable Systems Arising from Separation of Variables on $S^{3}$

TL;DR

This paper explores a family of integrable systems derived from orthogonal separation of variables on the sphere $S^3$, analyzing their degenerations, symmetries, and geometric structures using Poisson theory and symplectic reduction.

Contribution

It introduces a new family of integrable systems on $S^3$ and $S^2 imes S^2$, characterizing their degenerations, symmetries, and topological structures with a focus on global symplectic classification.

Findings

The family includes systems with $SO(2)$ and $SO(3)$ symmetries.

The momentum map's image forms an equilateral triangle, with degenerations leading to semi-toric and Delzant polygons.

The topology of all orthogonally separable coordinates on $S^3$ is the Stasheff polytope $K^4$.

Abstract

We show that the space of orthogonally separable coordinates on the sphere $S^3$ induces a natural family of integrable systems, which after symplectic reduction leads to a family of integrable systems on $S^2 \times S^2$. The generic member of the family corresponds to ellipsoidal coordinates. We use the theory of compatible Poisson structures to study the critical points and critical values of the momentum map. Interesting structure arises because the ellipsoidal coordinate system can degenerate in a variety of ways, and all possible orthogonally separable coordinate systems on $S^3$ (including degenerations) have the topology of the Stasheff polytope $K^4$, which is a pentagon. We describe how the generic integrable system degenerates, and how the appearance of global $SO(2)$ and $SO(3)$ symmetries is the main feature that organises the various degenerate systems. For the whole family we show that there is an action map whose image is an equilateral triangle. When higher symmetry is present, this triangle ``unfolds'' into a semi-toric polygon (when there is one global $S^1$-action) or a Delzant polygon (when there are two global $S^1$-actions). We believe that this family of integrable systems is a natural playground for theories of global symplectic classification of integrable systems.

Equations

$$E_{\mathbf{K}}=\sum _i\frac{1}{k!}{\mathbf{K}}^{i_1\dots i_k}{p}_{i_1}\dots p_{i_k}.$$

$$E_{[\mathbf{K},\mathbf{L}]}=\{E_{\mathbf{K}},E_{\mathbf{L}}{\}}_{(s_i,p_i)}$$

$$\frac{1}{g_{ii}}=\frac{\det ({\Omega }_i)}{\det (\Phi )},$$

$$\sum _i\frac{1}{2}(g^{-1}{)}_{ii}{\left(\frac{\partial W}{\partial s_i}\right)}^2=h.$$

$${\Phi }^{-1}\mathfrak{p}=\mathbf{\eta },$$

$$x_i^2=\frac{{\prod }_{j=1}^n(s_j-e_i)}{{\prod }_{k\ne i}(e_k-e_i)}.$$

$$\circ :S^m\times S^{k_1-1}\times \cdots \times S^{k_{m+1}-1}$$

$$(\mathbf{z},{\mathbf{w}}_1,\dots ,{\mathbf{w}}_{m+1})$$

$$C_{ij}=\{c_i,c_j{\}}_{T^{\ast }{\mathbb{R}}^4}.$$

$$\{f,g\}=\{f,g{\}}_{T^{\ast }{\mathbb{R}}^4}+\{f,c_i{\}}_{T^{\ast }{\mathbb{R}}^4}(C^{-1}{)}_{ij}\{c_j,g{\}}_{T^{\ast }{\mathbb{R}}^4}$$

$$B_{(\mathbf{x},\mathbf{y})}=\left(\begin{array}{cc}0& id-\mathbf{x}{\mathbf{x}}^t{\left|\mathbf{x}\right|}^{-2}\\ -id+\mathbf{x}{\mathbf{x}}^t{\left|\mathbf{x}\right|}^{-2}& -(\mathbf{x}{\mathbf{y}}^t-\mathbf{y}{\mathbf{x}}^t){\left|\mathbf{x}\right|}^{-2}\end{array}\right).$$

$$\{f,g\}:={\left(\nabla f\right)}^T{B}_{(\mathbf{x},\mathbf{y})}\nabla g$$

$$x_j^2$$

$$H=-2\sum _{i=1}^3\frac{{\prod }_{j=1}^4(s_i-e_j)}{{\prod }_{k\ne i}(s_i-s_k)}{p}_i^2.$$

$${\Phi }_{el}=\frac{1}{4}\left(\begin{array}{ccc}-\frac{s_1^2}{A(s_1)}& -\frac{s_1}{A(s_1)}& -\frac{1}{A(s_1)}\\ -\frac{s_2^2}{A(s_2)}& -\frac{s_2}{A(s_2)}& -\frac{1}{A(s_2)}\\ -\frac{s_2^2}{A(s_3)}& -\frac{s_1}{A(s_3)}& -\frac{1}{A(s_3)}\end{array}\right).$$

$${\eta }_0$$

$${\ell }_{ij}^2=\frac{x_i^2{x}_j^2}{4}{\left(\sum _{k=1}^3\left(-\frac{1}{s_k-e_i}+\frac{1}{s_k-e_j}\right)p_k\right)}^2$$

$${\eta }_0(\mathbf{L})$$

$$p_i^2=\frac{-R(s_i)}{4A(s_i)}$$

$$F_i=\sum _{j=1,j\ne i}^4\frac{{\ell }_{ij}^2}{e_i-e_j},$$

$$\sum _{i=1}^4\frac{F_i}{z-e_i}$$

$$T^{\ast }{S}^3/S^1{|}_{H=h}=U^{\ast }{S}^3/S^1\cong \widetilde{Gr}(2,4)$$

$$B_{\mathbf{L}}=\left(\begin{array}{cccccc}0& {\ell }_{23}& {\ell }_{24}& -{\ell }_{13}& -{\ell }_{14}& 0\\ -{\ell }_{23}& 0& {\ell }_{34}& {\ell }_{12}& 0& -{\ell }_{14}\\ -{\ell }_{24}& -{\ell }_{34}& 0& 0& {\ell }_{12}& {\ell }_{13}\\ {\ell }_{13}& -{\ell }_{12}& 0& 0& {\ell }_{34}& -{\ell }_{24}\\ {\ell }_{14}& 0& -{\ell }_{12}& -{\ell }_{34}& 0& {\ell }_{23}\\ 0& {\ell }_{14}& -{\ell }_{13}& {\ell }_{24}& -{\ell }_{23}& 0\end{array}\right)$$

$${\mathcal{C}}_1$$

$${\mathcal{C}}_2$$

$$\{f,g{\}}_{\mathbf{L}}=(\nabla f{)}^T{B}_{\mathbf{L}}\nabla g$$

$$X_1$$

$$X_2$$

$$X_3$$

$$B_{\mathbf{X},\mathbf{Y}}=\left(\begin{array}{cc}\hat{\mathbf{X}}& 0\\ 0& \hat{\mathbf{Y}}\end{array}\right)$$

$$\hat{\mathbf{v}}\mathbf{u}=\mathbf{v}\times \mathbf{u}$$

$$({\eta }_1,{\eta }_2)\mapsto (\alpha {\eta }_1+4\beta h,{\alpha }^2{\eta }_2+\alpha \beta {\eta }_1+2{\beta }^{2}h)$$

Full text

1 Introduction

Classifying and cataloguing integrable systems is an important unsolved problem. Only for particular classes of systems does a classification exist. It was shown by Atiyah-Guillemin-Sternberg [3, 20] that the image of the momentum map of a toric system is a convex polytope, called the momentum polytope. This polytope completely classifies the toric system up to an equivariant symplectomorphism. Delzant [12] gave an explicit construction of toric manifolds based on their momentum polytopes. Topological classification of Liouville foliations has been extensively studied by Bolsinov and Fomenko [6] and extended towards the orbital classification of integrable systems. Vũ Ngọc and Pelayo have extended the toric classification to semi-toric systems [35, 36]. While a complete theory of classification is still far out of reach, we’d like to investigate the possibility of extending the current theory to broader classes of integrable systems.

A very well studied class of integrable systems are superintegrable systems, see, e.g., [18]. A classification of superintegrable systems in two and three degrees of freedom has been achieved by Kalnins, Kress and Miller in a series of works [23, 24, 25, 26, 27]. In their book [28] they highlight the link between superintegrability and separation of variables. Superintegrable systems that are separable in multiple coordinate systems provide a rich source of integrable systems. This is because each distinct separable coordinate system gives rise to a Stäckel integrable system [4]. Separable coordinates on conformally flat spaces has been extensively studied by Kalnins and Miller in [29, 30, 33]. More recently, Schöbel studied the space of separable coordinates on the $n$-sphere as an algebraic variety [38, 40, 39]. While it is known [28] that all separable coordinates on the sphere can be obtained as appropriate limits of the general Jacobi ellipsoidal coordinates, Schöbel’s work formalises this by giving a topology to this space in the form of the Stasheff polytope. The inspiration for this paper was to establish a similar topology in the space of integrable systems that arise from separating the geodesic flow on $S^{3}$ in this family of coordinates. It is likely that similar constructions can be done for any superintegrable and multi-separable systems. In fact, the idea to use multi-separability to define interesting fibrations has been used in [13, 16] for the most fundamental systems of classical mechanics, the harmonic oscillator and the Kepler problem. To then use the periodic flow of the superintegrable system for reduction was first done in [11] and this paper is the natural continuation of that work where instead of a superintegrable system on $\mathbb{R}^{3}$ a superintegrable system on $S^{3}$ is the starting point.

Another motivation for our work are the recent studies [1, 2, 19, 22] of various integrable systems on the compact symplectic manifold $S^{2}\times S^{2}$. We will show that the symplectic reduction of the geodesic flow on $S^{3}$ results in a reduced system on a compact symplectic leaf of $\mathfrak{so}^{*}(4)$ that is diffeomorphic to $S^{2}\times S^{2}$. The $3$-degrees of freedom integrable systems on $T^{*}S^{3}$ obtained from separation of variables descend to $2$-degrees of freedom systems on $S^{2}\times S^{2}$ through this quotient. We will employ more recent techniques in the theory of compatible Poisson structures and bi-Hamiltonian systems [5, 7] to study these systems in detail. This will allow us to realise these systems as special restricted cases of the Manakov top [32, 42].

A somehow related question is the study of separable systems depending on parameters, foremost the geodesic flow on an ellipsoid [34]. In [10, 9] the geodesic flow on 3-dimensional ellipsoids with various sets of equal semi-major axes has been studied. It is astonishing how similar these system – degenerate or not – are to the ones studied in this paper. However, the fundamental difference is that there a constant energy slice of a 3-degree of freedom system is studied, while here we reduce and study the resulting 2-degree of freedom system. As a result, here we obtain a system on a compact symplectic manifold, which is better suited as a playground for symplectic classification. Similar degenerations have also been studied for the Neumann system [13] and again there are many similarities.

The paper is structured as follows. Section 2 introduces the basic theory of separation of variables. We focus on the separation of variables in the ellipsoidal coordinate system in Section 3. Section 4 discusses the symplectic reduction of the geodesic flow on $S^{3}$ with emphasis on establishing the reduced ellipsoidal integrable system on $S^{2}\times S^{2}$. The theory of compatible Poisson structures is applied to study the reduced ellipsoidal integrable system in Section 5. In Section 6, we combine the techniques and results of Sections 3, 4, and 5 to study the integrable systems obtained from separation of variables in the degenerate coordinate systems, namely prolate, oblate, Lamé, spherical and cylindrical coordinates.

2 Orthogonally Separable Coordinate Systems on $S^{3}$

In this section, we introduce some basic concepts from the theory of separation of variables. For more details, see [28, 4]. Let $(s_{i},p_{i})$ be local canonical coordinates on $T^{*}M$ where $M$ is an $n-$dimensional differentiable manifold with metric $g$. Define $\mathcal{S}^{k}(M)$ to be the space of smooth contravariant symmetric tensors of order $k$ on $M$, in particular $k=1$ is the space of vector fields on the manifold. Each $\bm{K}\in\mathcal{S}^{k}(M)$ can be associated with a $C^{\infty}$ real function $E_{\bm{K}}$ on $T^{*}M$ locally expressed in the momenta as

[TABLE]

The Lie bracket between two tensors $\bm{K}\in\mathcal{S}^{k}(M)$ and $\bm{R}\in\mathcal{S}^{r}(M)$, denoted by $[\bm{K},\bm{R}]\in\mathcal{S}^{k+r-1}(M)$, is defined by

[TABLE]

where $\{\cdot,\cdot\}_{(s_{i},p_{i})}$ denotes the canonical Poisson bracket in the curvilinear coordinates.

Definition 1**.**

On a Riemannian manifold $(M,g)$, a symmetric tensor $\bm{K}$ is a Killing tensor if it commutes with the metric $[\bm{K},g]=0$.

There is a natural identification of Killing tensors with first integrals of the geodesic flow: A function $E_{\bm{K}}$ is a first integral of the geodesic flow if and only if $\bm{K}$ is a Killing tensor, see, e.g [4]. Similarly to how a set of $n$ integrals in involution on a manifold form a Liouville-integrable system, a collection of $n$ Killing tensors defines a Stäckel system.

Definition 2**.**

A Stäckel system on an $n-$dimensional Riemannian manifold $(M,g)$ is a set of $n$ Killing tensors of order $2$ that commute under the commutator in (2).

Eisenhart proved in [17] that there is a bijective correspondence between equivalence classes of Stäckel systems and equivalence classes of orthogonal separable coordinate systems. Given an orthogonal coordinate system $s_{i}$ on an $n$-dimensional manifold $M$, we can define a Stäckel system by constructing a Stäckel matrix.

Definition 3**.**

A Stäckel matrix $\Phi$ for a given metric $g$ is any $n\times n$ matrix where each row depends on only one of the curvilinear coordinates $s_{i}$, and

[TABLE]

where $\Omega_{i}$ is the minor formed by deleting the $i^{th}$ row and first column of $\Phi$ and $g_{ii}$ is the $(i,i)$ element of the metric tensor.

The functional value of the Hamiltonian $H$ will be denoted by $h$. Let $W\coloneqq W(\bm{s},\bm{\eta})$ where $\bm{s}=(s_{1},\dots,s_{n})$ are the separable curvilinear coordinates and $\bm{\eta}=(\eta_{0},\dots,\eta_{n-1})$ are parameters. The Hamilton Jacobi equation is given by

[TABLE]

This can be separated by computing

[TABLE]

where $\mathfrak{p}=\left(p_{1}^{2},\dots,p_{n}^{2}\right)^{t}$ is the vector of squared canonical curvilinear momenta and the parameters $\bm{\eta}$ are the separation constants. Comparing (5) and (1), we see that rows of $\Phi^{-1}$ encode the diagonal entries of the Killing tensors for separable orthogonal coordinates $s_{i}$. The first row of (5) gives (4) and so we have $\eta_{0}=h$.

The work of Kalnins and Miller [29] gave a graphical algorithm for constructing all orthogonally separable coordinates on constant curvature manifolds. Recent results by Schöbel and Veselov extended this by giving an algebraic geometric classification of separable coordinate systems on $S^{n}$ [40, 38]. In particular, they showed that the variety of Stäckel systems on $S^{n}$ is given by the Stasheff polytope $K_{n+1}$ which is a convex polytope of dimension $n-1$. In this paper we work only with $S^{3}$; the relevant Stasheff polytope $K_{4}$ is shown in Figure 1.

The codimension [math] face of a Stasheff polytope represents the family of ellipsoidal coordinates $s_{i}$. These are defined as the roots of $T(s)=\sum_{i=1}^{n+1}\frac{x_{i}^{2}}{s-e_{i}}=0$ where $x_{i}$ are Cartesian coordinates on $\mathbb{R}^{n+1},$ $e_{j}\leq s_{j}\leq e_{j+1}$ for all $j=1,\dots,n$, the $e_{i}\geq 0$ are all distinct and are called the semi-major axes. Note that when using a similar coordinate system to separate the geodesic flow on an ellipsoid these parameters actually are semi-major axes (hence the name), while here they describe a separating coordinate system but not the underlying manifold. Solving $T(s_{j})=0$ together with $\sum_{i=1}^{n+1}x_{i}^{2}=1$ gives

[TABLE]

Note that for $S^{3}$, despite the ellipsoidal coordinates being parametrised by the 4 parameters $e_{1}<e_{2}<e_{3}<e_{4}$, the Stasheff polytope is only $2$-dimensional. Affine transformation of the parameters of the form $e_{i}\mapsto\alpha e_{i}+\beta$ for $\alpha\neq 0$ gives a equivalent coodinate system up to scaling. Thus each ellipsoidal coordinate system on $S^{3}$ with parameters $e_{1}<e_{2}<e_{3}<e_{4}$ can be transformed to an equivalent system with parameters $0<1<a<b$, see [28] for details.

Higher codimension faces represent families of degenerate coordinate systems on the sphere. Degenerate coordinate systems on $S^{n}$ are constructed by gluing ellipsoidal coordinates on lower dimensional spheres together. Let $\bm{w}_{\alpha}$ and $\bm{z}$ be Cartesian coordinates expressed in terms of local ellipsoidal coordinates on $S^{k_{\alpha}-1}$ and $S^{m}$ respectively where $\alpha=1,\dots,m+1$ and $n=k_{1}+\dots k_{m+1}-1$. Then any degenerate coordinate system on $S^{n}$ is found by recursively applying composition [40]

[TABLE]

where $z_{\beta}\bm{w}_{\beta}$ are Cartesian coordinates on $\mathbb{R}^{n+1}$ expressed in the new degenerate coordinate system.

In this paper we have chosen to adopt the notation of Schöbel [38]. The general ellipsoidal coordinates on $S^{n}$ are represented as $(1\ 2\ \dots\ n+1)$. If one attaches an $S^{j}$ to the $m^{\text{th}}$ Cartesian coordinate, then we enclose brackets around all numbers $m$ to $m+j$. For instance, attaching $S^{1}$ to the the $x_{2}$ coordinate on $S^{2}$ is written as $(1\ (2\ 3)\ 4)$. Symmetric bracketing results in systems with similar behaviour, i.e. $(1\ 2\ (3\ 4))$ and $((1\ 2)\ 3\ 4)$ describe equivalent coordinate systems as we will discuss in more detail below, also see [28, 39, 40, 38]. For the various degenerate coordinate systems we are going to use short hand names as indicated in Figure 1.

3 Separation of variables of the Geodesic Flow on $S^{3}$ and Ellipsoidal Coordinates

Consider the geodesic flow on the $3-$sphere as a constrained system on $T^{*}\mathbb{R}^{4}$ with Cartesian coordinates $(\bm{x},\bm{y})$ where $\bm{x}\cdot\bm{x}=1$ and $\bm{x}\cdot\bm{y}=0$. We define the Poisson bracket on $T^{*}S^{3}$ to be the Dirac bracket $\{\cdot,\cdot\}$ enforcing these two constraints. Let $\{\cdot,\cdot\}_{T^{*}\mathbb{R}^{4}}$ be the canonical Poisson bracket on $T^{*}\mathbb{R}^{4}$. Set $c_{1}=\bm{x}\cdot\bm{x},c_{2}=\bm{x}\cdot\bm{y}$ and define the matrix

[TABLE]

The Dirac bracket on $T^{*}S^{3}$ is given by

[TABLE]

with structure matrix

[TABLE]

From (7), the Poisson bracket between two functions $f$ and $g$ on $T^{*}S^{3}$ is

[TABLE]

where $\nabla$ denotes the gradient with respect to the Cartesian coordinates $(\bm{x},\bm{y})$.

Let $H=\frac{1}{2}\bm{y}\cdot\bm{y}$ be the Hamiltonian of the geodesic flow on $S^{3}$. It is well known that this system is superintegrable and separates in multiple coordinate systems [38]. The Hamilton Jacobi equation can be separated in general ellipsoidal coordinates $(s_{1},s_{2},s_{3})$ given by (6) for $n=3$ as

[TABLE]

where $0\leq e_{1}\leq s_{1}\leq e_{2}\leq s_{2}\leq e_{3}\leq s_{3}\leq e_{4}$ and the $e_{i}$’s are all distinct. In these coordinates, the geodesic Hamiltonian is (see e.g. [10, 28])

[TABLE]

This system is Liouville integrable. To separate the Hamilton Jacobi equation we use the following Stäckel matrix

[TABLE]

where $A(z)=\prod_{k=1}^{4}(z-e_{k})$. From (5), we compute $\Phi_{el}^{-1}\mathfrak{p}=(\eta_{0},-\eta_{1},\eta_{2})^{T}$ where

[TABLE]

and $D(s_{i})=\prod_{k\neq i}(s_{i}-s_{k}).$ To express (12) in terms of the angular momenta $\ell_{ij}\coloneqq x_{i}y_{j}-x_{j}y_{i}$, we note that

[TABLE]

with $x_{i}$ given by (9). Let $\bm{L}\coloneqq(\ell_{12},\ell_{13},\ell_{14},\ell_{23},\ell_{24},\ell_{34})$, it can be verified that

[TABLE]

The separated equations are obtained by multiplying both sides of (5) by $\Phi_{el}$. This gives

[TABLE]

where $R(z)=2hz^{2}-\eta_{1}^{*}z+\eta_{2}^{*}$ and $(\eta_{1}^{*},\eta_{2}^{*})$ are the values of the integrals $(\eta_{1},\eta_{2})$. Thus, the geodesic flow is separable on the hyperelliptic curve $w^{2}=-R(z)A(z)$ which has genus $3$.

It is known [34] that a set of global polynomial integrals for the geodesic flow are given by

[TABLE]

where $i\in\{1,2,3,4\}.$ The $F_{i}$ are known as the Uhlenbeck integrals and satisfy $\sum_{i=1}^{4}F_{i}=0$. The separation constants $\eta_{1}$ and $\eta_{2}$ are related to the $F_{i}$ via the identity

[TABLE]

Since all integrals are polynomial, wee can easily show that $\eta_{1},\eta_{2}$ and $H$ are functionally independent almost everywhere and $\{\eta_{1},H\}=\{\eta_{2},H\}=\{\eta_{1},\eta_{2}\}=0$. This establishes that the triple $(H,\eta_{1},\eta_{2})$ is an integrable system on $T^{*}S^{3}$. We call this the ellipsoidal integrable system on $T^{*}S^{3}$. This is a reformulation of the underlying Stäckel system, whose commuting Killing tensors lead to quadratic (in momenta) integrals, which are also quadratic in angular momenta.

Since $H$ is a global $S^{1}$ action on the energy surface where $2h=1$ we can use it to perform symplectic reduction. Doing so, we obtain a reduced system on the symplectic manifold $S^{2}\times S^{2}$. The integrals $(\eta_{1},\eta_{2})$ descend to form an integrable system on this quotient space.

4 Reduction by Geodesic Flow

The orbits of $H$ are oriented great circles on $S^{3}$. Since each great circle is the intersection of a two dimensional plane through the origin with $S^{3}$, the orbit space of $H$ is given by

[TABLE]

where the oriented Grassmanian $\widetilde{Gr}(2,4)$ is the set of oriented two dimensional planes in $\mathbb{R}^{4}$ and $U^{*}S^{3}$ is the unit cotangent bundle of $S^{3}$. One way to see that $\widetilde{Gr}(2,4)\cong S^{2}\times S^{2}$ is using the Plücker embedding (for more details on this, see Appendix A.1). For our purposes, the reduction will be performed with invariants of the geodesic Hamiltonian.

Invariants of $H$ are the six angular momenta $\bm{L}\coloneqq(\ell_{12},\ell_{13},\ell_{14},\ell_{23},\ell_{24},\ell_{34})$. These form a closed set of invariants under the Dirac bracket $\{\cdot,\cdot\}$ from (8). The Poisson algebra of these invariants has the structure matrix

[TABLE]

with 2 Casimirs: $\mathcal{C}_{1}=2H=\sum_{j>i}\ell_{ij}^{2}$ and $\mathcal{C}_{2}=\ell_{12}\ell_{34}-\ell_{13}\ell_{24}+\ell_{14}\ell_{23}$. The first is the energy of the geodesic flow which we have the freedom to set to an arbitrary value $2h$. The second Casimir is the Plücker relation and must be zero since the angular momenta $\bm{L}=\bm{x}\wedge\bm{y}$ where $\wedge$ is the wedge operator. This means that $\bm{L}$ is a totally decomposable bivector and so must satisfy $C_{2}=\left|\bm{L}\wedge\bm{L}\right|=0$. The Lie Poisson algebra of the $\ell_{ij}$’s is isomorphic to the Lie algebra $\mathfrak{so}(4)$.

Using the $\ell_{ij}$ as new coordinates, we obtain an explicit realisation of $S^{2}\times S^{2}$ as

[TABLE]

The Poisson bracket of functions on $S^{2}\times S^{2}$, denoted by $\{\cdot,\cdot\}_{\bm{L}}$ is

[TABLE]

where $\nabla$ denotes the gradient with respect to $\bm{L}$.

A sometimes more convenient set of coordinates on $S^{2}\times S^{2}$ is obtained by applying the linear transformation $T:\bm{L}\leavevmode\nobreak\ \mapsto(\bm{X},\bm{Y})=\leavevmode\nobreak\ (X_{1},X_{2},X_{3},Y_{1},Y_{2},Y_{3})$ with

[TABLE]

In these variables we can rewrite (18) as $\mathscr{C}_{1}=4\left|\bm{X}\right|^{2}$ and $\mathscr{C}_{2}=4\left|\bm{Y}\right|^{2}$ which both have functional value $2h$. The Poisson structure (7) becomes block diagonal

[TABLE]

and is isomorphic to $\mathfrak{so}(3)\times\mathfrak{so}(3)$. The notation in (21) is such that for a vector $\bm{v}\in\mathbb{R}^{3}$ the corresponding antisymmetric hat matrix $\hat{\bm{v}}$ is defined by

[TABLE]

This reduction gives an integrable system on the reduced manifold $S^{2}\times S^{2}$. While the integrals $(\eta_{1},\eta_{2})$ can be easily rewritten in terms of the $(\bm{X},\bm{Y})$ variables using (20), they are simplest and most symmetric as functions of the $\ell_{ij}$’s as in (13).

Under (19), $(\eta_{1},\eta_{2})$ are commuting quadratic functions on $S^{2}\times S^{2}$ and so we arrive at the following result.

Theorem 1**.**

The integrable sytem $(H,\eta_{1},\eta_{2})$ on $T^{*}S^{3}$ descends to an integrable system $(\eta_{1}(\bm{L}),\eta_{2}(\bm{L}),\{\cdot,\cdot\}_{\bm{L}})$ on $S^{2}\times S^{2}$ with two degrees of freedom and integrals quadratic in $\ell_{ij}$.

We call this integrable system the reduced ellipsoidal integrable system. The symplectic reduction performed in this section also applies to integrable systems obtained from separating the geodesic flow in the degenerate coordinate systems shown in Figure 1.

Separating coordinate systems on $S^{3}$ are invariant under affine transformations $e_{i}\mapsto\alpha e_{i}+\beta$ for $\alpha\neq 0$. This allows us to normalise the ordered distinct parameters $(e_{1},e_{2},e_{3},e_{4})$ to $(0,1,a,b)$ by a shift by $\beta=-e_{1}$ and a scaling by $\alpha=e_{2}-e_{1}$. Thus the inside of Figure 1 can be thought of as the region $1<a<b$. The affine transformations when applied to the family of corresponding integrable systems on $T^{*}S^{3}$ gives topologically equivalent integrable systems. This property can be observed directly in the reduced systems on $S^{2}\times S^{2}$.

Lemma 1**.**

Affine transformation of the parameters $e_{i}\mapsto\alpha e_{i}+\beta$ for $\alpha\neq 0$ when applied to the reduced system $(\eta_{1}(\bm{L}),\eta_{2}(\bm{L}))$ gives a topologically equivalent integrable system.

Proof.

Applying $e_{i}\mapsto\alpha e_{i}+\beta$ to $(\eta_{1}(\bm{L}),\eta_{2}(\bm{L}))$ induces the map

[TABLE]

which gives a topologically equivalent system, because it is a linear map of the original integrals, plus affine terms that add the Casimir $h$. ∎

This result illustrates nicely how the equivalence of separating coordinates leads to an equivalence of reduced integrable systems. It highlights the fact that the reduced system does not have a Hamiltonian (since we reduced by the flow of $H$) and hence it is natural to consider quadratic integrals up to linear transformations.

There is another equivalence between separating coordinates which maps an ordered quadruple $(e_{1},e_{2},e_{3},e_{4})$ to an ordered quadruple $(-e_{4},-e_{3},-e_{2},-e_{1})$ . After normalisation this maps $(0,1,a,b)$ to $(0,1,a^{\prime},b^{\prime})$ where $a^{\prime}=(b-1)/(b-a)$ and $b^{\prime}=b/(b-a)$. This map is an involution that can be written as

[TABLE]

The line of fixed points of the involution is $1/b+a/b=1$. This suggests to map the parameter region $1<a<b$ to the triangle $0<a/b<1/b<1$ which is cut in half by the line of fixed points, see Figure 2 a). The prolate case $a=1$ and the oblate case $a=b$ correspond to two edges of this triangle. To see the whole parameter space the $e_{i}$ need to be considered projectively. In particular the point $1=a=b$ representing the Lamé family needs to be blown up.

Lemma 2**.**

Define $q=1/b,r=(a-1)/(b-1)$. There is a one-to-one correspondence between equivalence classes of reduced integrable system on $S^{2}\times S^{2}$ and points in the region $0\leq q\leq 1$, $0\leq r\leq 1$, $q\geq(1-r)/(2-r)$.

Proof.

The map $r=(a-1)/(b-1)$ blows up the point $1=a=b$ to a line. The parameter $r$ is the inverse of the slope of a straight line in $ab$-space through the point $a=b=1$. Due to $1\leq a\leq b$ we have $0\leq r\leq 1$. Consider the images of the edges of the triangle $0<a/b<1/b<1$. The prolate line segment is mapped to $r=0$, $1/2<q<1$. The oblate line segment is mapped to $r=1$, $0<q<1$. The line of fixed points is mapped to $q=(1-r)/(2-r)$. This establishes the claimed boundaries of the region. The corners of the region are:

• •

$(q,r)=(0,1/2)$: symmetric prolate coordinates.

• •

$(q,r)=(0,1)$: spherical coordinates.

• •

$(q,r)=(1,1)$: spherical coordinates.

• •

$(q,r)=(0,1)$: cylindrical coordinates.

∎

The region described in the Lemma in $(r,q)$ space is shown in Figure 2 b). It represents half of the Stasheff polytope Figure 1.

In the next section, we study the reduced ellipsoidal system in depth. We find the momentum map, compute the critical points and critical values. The integrable systems arising from the degenerate coordinate systems will be covered in detail in section 6.

5 The Reduced Ellipsoidal Integrable System

To construct and study the bifurcation diagram for the ellipsoidal integrable system we will employ techniques from [5] and [7] using compatible Poisson structures.

5.1 Compatible Poisson Structures

In this section, we will be closely following Example B in [7]. Let us consider the reduced system on a symplectic leaf of $\mathfrak{so}^{*}(4)$ defined by $\mathcal{C}_{1}=\bm{L}\cdot\bm{L}=2h$ and $\mathcal{C}_{2}=\ell_{12}\ell_{34}-\ell_{13}\ell_{24}+\ell_{14}\ell_{23}=0$. On $\mathfrak{so}(4)$ we have the standard bracket $[X,Y]=XY-YX$ and we can identify elements $X\in\mathfrak{so}(4)$ with elements $X^{*}\in\mathfrak{so}^{*}(4)$ via

[TABLE]

Here $K$ is the Killing form defined as

[TABLE]

where $\text{ad}_{X}=[X,\cdot]$. Explicitly, let us define $X_{ij}$ to be the $4\times 4$ matrix with $1$ in the $ij^{\text{th}}$ position, $-1$ in the $ji^{\text{th}}$ position and [math] everywhere else. This gives us a basis of $\mathfrak{so}(4)$. An element $X\in\mathfrak{so}(4)$ of the form

[TABLE]

can be written as $X=\sum_{j>i}\ell_{ij}X_{ij}$. This allows for the further identification of $T(\mathfrak{so}^{*}(4))\equiv\mathfrak{so}(4)$ with $\nabla_{\bm{L}}f\leftrightarrow\sum_{i<j}\frac{\partial_{f}}{\partial\ell_{ij}}X_{ij}$. We can now express the integrals $\eta_{1}$ and $\eta_{2}$ as functions on $\mathfrak{so}(4)$ as

[TABLE]

with $\nabla_{\bm{L}}\eta_{1}\coloneqq A_{1}=\sum_{i<j}(e_{m}+e_{n})\ell_{ij}X_{ij}$ and $\nabla_{\bm{L}}\eta_{2}\coloneqq A_{2}=\sum_{i<j}e_{m}e_{n}\ell_{ij}X_{ij}$ where $i,j,m,n$ are all distinct. Dynamics on $\mathfrak{so}^{*}(4)$ with Hamiltonian $\eta_{1}$ or $\eta_{2}$ can be rewritten in Lax form as $\dot{X}=[A_{1},X]$ or $\dot{X}=[A_{2},X]$. The trace of $X^{2}$ and $X^{4}$ recover the Casimirs $-2\mathcal{C}_{1}$ and $-4\mathcal{C}_{2}^{2}+2\mathcal{C}_{1}^{2}$, respectively.

Let $C$ be a symmetric matrix and define the Lie bracket $[X,Y]_{C}\coloneqq XCY-YCX$. This lifts to the Poisson bracket $\{\cdot,\cdot\}_{C}$ on $\mathfrak{so}^{*}(4)$. We can WOLG assume that $C$ is diagonal of the form $C=\text{diag}(c_{1},c_{2},c_{3},c_{4})$. If $C$ is invertible, then there exists an isomorphism $\alpha$ from $\mathfrak{so}(4,\mathbb{C})$ to $\mathfrak{so}(4,[\cdot,\cdot]_{C})$ defined by $\alpha:X\mapsto C^{1/2}XC^{1/2}$. This lifts to a linear map

[TABLE]

where $M_{C}=\text{diag}(\sqrt{c_{1}c_{2}},\sqrt{c_{1}c_{3}},\sqrt{c_{1}c_{4}},\sqrt{c_{2}c_{3}},\sqrt{c_{2}c_{4}},\sqrt{c_{3}c_{4}}).$

The Poisson matrix for $\mathfrak{so}^{*}(4,\{\cdot,\cdot\}_{C})$ in the basis of $(\ell_{12},\ell_{13},\ell_{14},\ell_{23},\ell_{24},\ell_{34})$ is given by

[TABLE]

In [7] it is shown that the Lie bundle $\lambda[\cdot,\cdot]-[\cdot,\cdot]_{C}=[\cdot,\cdot]_{\lambda I-C}$ where $\lambda\in\mathbb{R}\cup\infty$ is still a Lie bracket on $\mathfrak{so}(4)$. Similarly, $\lambda\{\cdot,\cdot\}-\{\cdot,\cdot\}_{C}=\{\cdot,\cdot\}_{\lambda I-C}$ is also a Poisson bracket on $\mathfrak{so}^{*}(4)$ giving us a set of compatible Poisson structures. Following Example B in [7], we can now study our system from the perspective of the compatible Poisson structures $\{\cdot,\cdot\}_{\lambda I-C}$ on $\mathfrak{so}^{*}(4)$. Expanding the Casimirs in terms of the parameter $\lambda$ gives commuting integrals [7].

Proposition 3** ([7]).**

The integrals $(I_{0},I_{1},I_{2})$ for the Poisson structure $(\mathfrak{so}^{*}(4),\{\cdot,\cdot\}_{\lambda I-C})$ with $C=\text{diag}(c_{1},c_{2},c_{3},c_{4})$ where $c_{i}$ are real distinct constants can be obtained from the coefficients of the numerator of the rational function given by

[TABLE]

where $X\in\mathfrak{so}(4)$. They are $I_{0}=-\sum_{i,j}\ell_{ij}^{2}=-2h,\ I_{1}=\sum_{i<j}(c_{n}+c_{m})\ell_{ij}^{2},\ I_{2}=-\sum_{i<j}c_{n}c_{m}\ell_{ij}^{2}$ where the indices $m,n,i,j$ are all distinct.

Define $E=\text{diag}(e_{1},e_{2},e_{3},e_{4})$ and using $C=E$ gives us the integrals $(I_{0},I_{1},I_{2})=(-2h,\eta_{1},-\eta_{2})$ obtained from separation of variables in (13). This allows us to study the ellipsoidal integrable system on the reduced space $\mathfrak{so}^{*}(4)$ as a system of compatible Poisson structures and (24) becomes

[TABLE]

This is precisely the equation for the separated momenta in (14) with $\psi(s_{i})=4p_{i}^{2}$.

5.2 Critical Points

To find critical points, consider the lift of the standard endomorphism from $\mathfrak{so}(4,\mathbb{C})$ to $\mathfrak{so}(3)\oplus\mathfrak{so}(3)$ defined by T:\text{\bm{L}\mapsto(\bm{X},\bm{Y})} given in (20), where here $\bm{X}$ and $\bm{Y}$ are complex vectors. We will be using the map $\gamma_{\lambda I-E}$ in (23) to construct the map $T_{2}=\gamma_{\lambda I-E}T^{-1}$ from $[\mathfrak{so}(3)\oplus\mathfrak{so}(3)]^{*}$ to $\mathfrak{so}^{*}(4,\{\cdot,\cdot\}_{C})$. It is known from [7] that the set of singular points in $\mathfrak{so}(4,\mathbb{C})\equiv\mathfrak{so}(3)\oplus\mathfrak{so}(3)$ under the standard bracket is given by $\{(\bm{X},0)\}\cup\{(0,\bm{Y})\}$. When the matrix $\lambda I-E$ is invertible (that is $\lambda\neq e_{i}$), the map $\gamma_{\lambda I-E}$ is an Poisson isomorphism between $\mathfrak{so}^{*}(4,\mathbb{C})$ and $\mathfrak{so}^{*}(4,\{\cdot,\cdot\}_{\lambda I-E})$ and so $T_{2}$ is also a Poisson isomorphism. The set of singular points of $\mathfrak{so}^{*}(4,\{\cdot,\cdot\}_{\lambda I-E})$ is the image of the singular points of $\mathfrak{so}(4,\mathbb{C})$ under $T_{2}$, that is $T_{2}(\{(\bm{X},0)\}\cup\{(0,\bm{Y})\})$. The set of critical points of the ellipsoidal integrable system is precisely the set of singular points of $\mathfrak{so}^{*}(4,\{\cdot,\cdot\}_{\lambda I-E})$ by Theorem 2 in [7]. We have an analogous result:

Proposition 4**.**

An element of $\bm{L}\in\mathfrak{so}^{*}(4)$ is critical if

1. $\lambda\neq e_{i}$ and $\bm{L}\in\Re e(T_{2}(0,\bm{X})\cup T_{2}(\bm{\bm{Y}},0))$.

2. $\lambda=e_{i}$ and $\bm{L}$ is such that the Poisson bracket $\{\cdot,\cdot\}_{C}$ drops rank.

Using Proposition 4 we start by finding the general solutions for case 1 with $\lambda\neq e_{i}$. The critical points are given by $\bm{L}\in\Re e(T_{2}(0,\bm{Y})\cup T_{2}(\bm{\bm{X}},0))$. Let $\bm{z}=(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}$, define $\text{Sing}_{+}=\{(\bm{\bm{z}},0)\},\ \text{Sing}_{-}=\{(0,\bm{\bm{z}})$} and $\text{Sing}=\text{Sing}_{+}\cup\text{Sing}_{-}$. We have

[TABLE]

where $z_{j}=a_{j}+ib_{j}$ with $a_{i},b_{i}\in\mathbb{R}$. Substituting these into the integrals gives $(I_{1},I_{2})=(4h\lambda,2h\lambda^{2})$ which is the curve $I_{2}=\frac{I_{1}^{2}}{8h}$ . This is precisely when $\lambda$ is a double root of (25). It is easily seen that the values of $\lambda$ that permit a double root while keeping $\psi>0$ are in the interval $\lambda\in[e_{2},e_{3}]$. After taking the real part this gives the critical points

[TABLE]

We can verify that these are critical points of the system $(\eta_{1},\eta_{2})$ by noting that $B(\nabla\eta_{0}-\lambda\nabla\eta_{1}-\nabla\eta_{2})|_{\bm{L=L_{\pm}}}=0$. Substituting (27) into the Plücker relation forces

[TABLE]

Using $\bm{L\cdot L}=2h$ gives the conic

[TABLE]

The conditions (28) and (29) when combined with (27) gives the explicit parametrisation for the 4 topological $S^{1}$ of critical points for case 1 with $\lambda\neq e_{i}$.

In case 2 where $\lambda=e_{i}$, the map $\gamma_{\lambda I-E}$ still exists but it is not invertible and so $T_{2}(e_{i})(\text{Sing})$ is still a subset of the critical points of the bracket $\{\cdot,\cdot\}_{C(e_{i})}$. Let us consider $\lambda=e_{1}$, then we have

[TABLE]

These naturally satisfies the Plücker relations giving us solutions of the form $\bm{L}=(0,0,0,\ell_{23},\ell_{24},\ell_{34})$ after taking the real part with constraint $\ell_{23}^{2}+\ell_{24}^{2}+\ell_{34}^{2}=2h$. This means that the set of all critical points corresponding to case 2 with $\lambda=e_{1}$ is the sphere $\ell_{23}^{2}+\ell_{24}^{2}+\ell_{34}^{2}=2h$.

In order to show these are all the critical points for $\lambda=e_{1}$, recall that the singular points of $\{\cdot,\cdot\}_{C(e_{i})}$ occurs when $B_{C(e_{i})}$ drops rank. We perform the change of variables $(\bm{U},\bm{V})$ where $\bm{U}=(u_{12},u_{13,},u_{14})$ and $\bm{V}=(v_{34},v_{24},v_{23})$ with $u_{ij}=\sqrt{(e_{i}-e_{k})(e_{i}-e_{m})}\ell_{ij}$ , $v_{km}=\frac{1}{\sqrt{(e_{i}-e_{k})(e_{i}-e_{m})}}\ell_{km}$ and $i,j,k,m$ are all distinct. This transforms $B_{e_{1}I-E}$ into the standard $\mathfrak{e}^{*}(3)$ algebra given by

[TABLE]

Singular orbits of $B_{3}$ are given by $\bm{U}=\bm{0}$, giving $\ell_{12}=\ell_{13}=\ell_{14}=0$ for all $j\neq i$. These are the critical points described above.

For $\lambda=e_{2},e_{3},e_{4}$ we have an isomorphism between $B_{\lambda I-E}$ and $\mathfrak{e}^{*}(1,2),\ \mathfrak{e}^{*}(2,1),\ \mathfrak{e}^{*}(0,3)$ respectively, all of which have singular orbits iff $\bm{U}=\bm{0}$ giving $\ell_{ik}=0$ for all $k\neq i$ if $\lambda=e_{i}$.

5.3 Bifurcation diagram

Using the critical points described in the previous section we have the following result for the critical values.

Corolary 5**.**

The critical values of the integrals $(\eta_{1},\eta_{2})$ occur when $\lambda$ is a real root of $\psi(\lambda)$ so that $\psi(\lambda)\geq 0$ and

$\lambda=e_{i}$* or* 2. 2.

$\lambda$* is a double root of the numerator $\psi(\lambda)\coloneqq 2h\lambda^{2}-\eta_{1}\lambda+\eta_{2}$*

Proof.

By the Cayley Hamilton theorem, it is known that

[TABLE]

We observe that if $M=XC^{-1}$, $\text{Tr}(M)=0$ and $\det(M)=\frac{(X_{12}X_{23}-X_{13}X_{24}+X_{12}X_{34})^{2}}{(\lambda-c_{1})(\lambda-c_{2})(\lambda-c_{3})(\lambda-c_{4})}=\frac{\mathcal{C}_{2}^{2}}{(\lambda-c_{1})(\lambda-c_{2})(\lambda-c_{3})(\lambda-c_{4})}=0$ due to constraint on the Casimir. This implies that $\lambda$ has to be a root of $\psi(\lambda)$ for valid motion. For critical points, we either have $\lambda=e_{i}$ or $\lambda$ such that $(I_{1},I_{2})=(4h\lambda,2h\lambda^{2})$, that is $\lambda$ is a double root of $\psi(\lambda)$. ∎

Proposition 6**.**

The set of critical values for the reduced ellipsoidal integrable system $(\eta_{1},\eta_{2}):S^{2}\times S^{2}\to\mathbb{R}^{2}$ is composed of 4 straight lines and a quadratic curve. The lines are $\mathcal{L}_{i}:\eta_{2}-e_{i}(\eta_{1}-e_{i})=0$ for $i\in\{1,2,3,4\}$ and part of the parabola $\eta_{2}=\frac{\eta_{1}^{2}}{4}$ given by $\mathcal{C}:(\eta_{1},\eta_{2})=\left(2t,t^{2}\right)$ for $e_{2}\leq t\leq e_{3}$. There are $6$ transverse intersections of the lines $\mathcal{L}_{i}\cap\mathcal{L}_{j}$ which occur at $(\eta_{1},\eta_{2})=d_{ij}\coloneqq(e_{i}+e_{j},e_{i}e_{j})$ where $i\neq j$ and $i,j\in\{1,2,3,4\}$. The points $d_{i}\coloneqq(2e_{i},e_{i}^{2})$ where $i\in\{2,3\}$ correspond to the two tangential intersections of $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ with $\mathcal{C}$. The bifurcation diagram with $2h=1$ is shown in Figure 3 b).

Proof.

Using Proposition 3 and Corollary 5, when $\lambda=e_{i}$ in (25) we must have $\eta_{2}=e_{i}(\eta_{1}-2he_{i})$ for the numerator of $\psi(\lambda)$ to be identically [math] . If $\lambda$ is a double root, then taking the discriminant of the numerator gives the curve $\eta_{2}=\frac{\eta_{1}^{2}}{8h}$. With $2h=1$, we obtain the formulae for the lines $\mathcal{L}_{i}$ and the curve $\mathcal{C}$ which make up the boundary of the image of the momentum map. Since the bifurcation diagram is necessarily compact we must also determine the regions for which the momenta are real. To do this, recall that $\psi(s_{i})=4p_{i}^{2}$ can be factored as follows

[TABLE]

where $e_{1}\leq r_{1}\leq r_{2}\leq e_{4}$, $\eta_{1}=r_{1}+r_{2}$ and $\eta_{2}=r_{1}r_{2}.$ The denominator of (30) defines $4$ poles at $e_{j}$ and so divides the interval $[e_{1},e_{4}]$ into three intervals $[e_{i},e_{i+1}]$ where $i\in\{1,2,3\}$. To distribute the roots $r_{k},$ we require that $p_{i}^{2}$ takes on non negative values in each interval $[e_{i},e_{i+1}]$ for valid motion. This gives $4$ regions of motion which we represent in Figure 3 a). We call this the root diagram for the ellipsoidal system. The mapping from the root diagram to the bifurcation diagram is smooth on the interior and all edges of the root diagrams except on the diagonal cyan segment where $r_{1}=r_{2}$. The image of the momentum map is the region enclosed by the lines $\mathcal{L}_{i}$ and the curve $\mathcal{C}$ presented in Figure 3 b).

From the root diagram, we find that each of the lines $\mathcal{L}_{i}$ is defined over $\eta_{1}\in[e_{1}+e_{j},e_{4}+e_{k}],\eta_{2}\in[e_{1}e_{j},e_{4}e_{k}]$ where $j=\max(2,i)$ and $k=\min(3,i)$. Hence the end points of $\mathcal{L}_{i}$ are $d_{ij}$ and $d_{k4}=d_{4k}$. The critical points on each line $\lambda=e_{i}$ described in the previous section represents the geodeosic subflow on the great $2$-sphere $x_{i}=0$ under elliptical coordinates on $S^{2}$ with axes given by the remaining $e_{k}$ with $k\neq i$. Consider the case with $\lambda=e_{1}$, for each critical value $(\eta_{1},\eta_{2})$ on the line $\mathcal{L}_{1}$, its set of critical points is the intersection of the sphere $\ell_{23}^{2}+\ell_{24}^{2}+\ell_{34}^{2}=2h$ with the ellipsoids $\eta_{2}(\bm{L})=e_{1}(e_{4}\ell_{23}^{2}+e_{3}\ell_{24}^{2}+e_{2}\ell_{34}^{2})=\eta_{2}$. These are precisely the fibres of the geodesic flow on $S^{2}$ when separation of variables is performed in the elliptical coordinates on $S^{2}$ with semi-axes $(e_{2},e_{3},e_{4})$ (see Appendix A.2.1). Indeed, when $\ell_{12}=\ell_{13}=\ell_{14}=0$, we have $x_{1}=y_{1}=0$ and we have geodesic motion restricted on the great 2-sphere $x_{1}=0$.

In the case where there is a double root in the numerator, i.e. $t=r_{1}=r_{2}$, we obtain the curve $\mathcal{C}:(\eta_{1},\eta_{2})=(2t,t^{2})$ where $e_{2}\leq t\leq e_{3}$.

It is clear that the intersections between $\mathcal{L}_{i}$ and $\mathcal{L}_{j}$ are transverse and are located at $(\eta_{1},\eta_{2})=(e_{i}+e_{j},e_{i}e_{j})=d_{ij}$ where $i\neq j$ and $i,j\in\{1,2,3,4\}$. Similarly, it is easy to see by computing the tangents that only $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ intersect $\mathcal{C}$ tangentially at $d_{2}=(2e_{2},e_{2}^{2})$ and $d_{3}=(2e_{3},e_{3}^{2})$ respectively. ∎

Corolary 7**.**

The Uhlenbeck integral $F_{i}=0$ if and only if $\eta_{2}-e_{i}(\eta_{1}-e_{i})=0$, i.e. $F_{i}$ vanishes along $\mathcal{L}_{i}$.

Proof.

Taking the residue at $e_{i}$ of both sides of (16) gives

[TABLE]

where indicies $i,k,l,m$ are all distinct. Since we have assumed all semi major axes are distinct, $F_{i}=0$ if and only if $(\eta_{1},\eta_{2})$ lie on $\mathcal{L}_{i}$. ∎

To classify the nature of the critical points, we compute the eigenvalues of the linearisation $\nabla[B(\lambda\nabla\eta_{1}+\nabla\eta_{2})]$. The intersections $d_{12},d_{14}$ and $d_{34}$ are all elliptic-elliptic critical values, $d_{13}$ and $d_{24}$ are of elliptic-hyperbolic type and $d_{23}$ is hyperbolic-hyperbolic. The tangential intersections $d_{2}$ and $d_{3}$ are degenerate. The lines $\mathcal{L}_{1},\mathcal{L}_{4}$, the curve $\mathcal{C}$, as well as the yellow and purple parts of $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ respectively have one pair of imaginary eigenvalues and so are codimension one elliptic. The magenta, orange and grey, red segments of $\mathcal{L}_{3}$ and $\mathcal{L}_{2}$ give one pair of real eigenvalues and so are codimension one hyperbolic.

5.4 Critical Fibres

Unlike the critical points, the parametrisation of the critical fibre cannot be computed algebraically. We will instead provide informal description and topological classification of the fibres on $S^{2}\times S^{2}$ instead.

Firstly, by Louvile-Arnold theorem, the preimage of regular values in regions $I,II,III,IV$ are $\mathbb{T}^{2}$ in $S^{2}\times S^{2}$.

Since $d_{12},d_{14},d_{34}$ are elliptic-elliptic, their preimage on $S^{2}\times S^{2}$ are 2 points each, the critical points found earlier.

Next, consider the lines immediately connected to $d_{12},d_{14},d_{34}$ These are the lines $\mathcal{L}_{1},\mathcal{L}_{4}$, as well as the yellow and purple parts of $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$ respectively. They are codimension one elliptic and so their fibres are circles $S^{1}$ and only contain critical points. The multiplicity of these circles is two as a result of extending the multiplicities of $d_{12},d_{14}$ and $d_{34}$.

Similarly, $d_{13}$ and $d_{24}$ are elliptic-hyperbolic critical values. There are 2 intersecting circles of critical points in their fibres. To obtain the full fibre, we observe that as we move along $\mathcal{L}_{1}$ (resp. $\mathcal{L}_{4})$ and pass by $d_{13}$ (resp. $d_{24})$, two $S^{1}$ bifurcate into two $S^{1}$. Such a bifurcation is represented by the Fomenko atom $C_{2}$.

The magenta and grey segments of $\mathcal{L}_{3}$ and $\mathcal{L}_{2}$ are extensions of $d_{13}$ and $d_{24}$ respectively, the fibres of these segments are $S^{1}\times C_{2}$.

Since the grey and magneta lines both have a $C_{2}$ type singularity, we know that the fibre of $d_{23}$ is of type $(C_{2},C_{2})$ $l$-type of complexity $2$ with loop molecule number 17 in [6]. The critical fibre is simpler, there are only 4 types after symmetry reduction [14].

Since the $17$ saddle saddle singularity contains $4$ $B$ atoms, it follows that the fibres of the red and orange lines are $2B\times S^{1}$.

The cyan curve is codimension one elliptic and so its fibre is a circle $S^{1}$. To find the multiplicity of these $S^{1}$, we note that the fibre of $d_{23}$ contains $4S^{1}$. Extending $d_{23}$ into chamber $IV$, we multiply by $S^{1}$. Hence, the fibre of a regular value in chamber $IV$ is $4T^{2}$. Continuing onto $\mathcal{C}$, we see that the fibre along the curve must be $4S^{1}$.

The fibre type does not change at the degenerate points $d_{2}$ and $d_{3}$, hence are simply $2S^{1}$. Approaching $d_{2}$ and $d_{3}$ along the yellow and purple lines respectively, we see $2S^{1}$ bifurcate into $4S^{1}$ (along the curve $\mathcal{C}$) as well as a hyperbolic fibre ($2B\times S^{1}$). This means $d_{2}$ and $d_{3}$ are pitchfork bifurcations as described in [6].

Extending the codimension $1$ lines $\mathcal{L}_{1}$ and $\mathcal{L}_{4}$ into chambers $I-III$ gives the following corollary.

Corolary 8**.**

The fibre of a regular point on the momentum map is a torus $T^{2}$. The multiplicity of the tori in chambers $I-III$ is $2$, while tori in chamber $IV$ have multiplicity $4$.

5.5 The Action Map

Recall that the Liouville tori of the ellipsoidal integrable system are certain coverings of the real parts of the Jacobi variety of the genus 3 hyperelliptic curve defined by $w^{2}(z)=-R(z)A(z)$. The actions of this system are the periods of the Abelian integral

[TABLE]

The actions $I_{j}$ are discontinuous on phase space as we cross boundaries of chambers of the bifurcation diagram. To construct a set of continuous actions, we perform discrete symmetry reduction by the $2^{4}$ discrete symmetries generated by the reflections

[TABLE]

Following [21], we can construct symmetry reduced actions that are continuous accross all chambers of the momentum map. A detailed explanation can be found in [15].

Lemma 9**.**

The continuous actions of the ellipsoidal integrable system $(J_{1},J_{2},J_{3})$ are

[TABLE]

This discrete symmetry reduction reduces the multiplicities of $T^{2}$ in all chambers to $1$ and this is the reason why the discrete symmetry reduced system has a globally continuous action map. The actions $J_{i}$ are independent on $T^{*}S^{3}$ but since $H$ is a superintegrable Hamiltonian, they are related on an energy surface, and hence become dependent for the reduced system on $S^{2}\times S^{2}$.

Lemma 10**.**

The continuous actions satisfy the relation

[TABLE]

Proof.

Let $\beta_{i}$ be cycles that enclose the bounds of $J_{i}$ respectively. I.e. $\beta_{1}$ encloses the interval $[e_{1},\min(r_{1},e_{2})]$ and similarly for $\beta_{2},\beta_{3}$. By deforming the cycles on the hyperelliptic curve $w^{2}=-R(z)A(z)$ we have

[TABLE]

where $\gamma$ is a cycle that encircles the point at infinity. It is easily shown that

[TABLE]

Combining (34) with (33) gives the desired result. ∎

Theorem 2**.**

The image of the action map (31) is an equilateral triangle $\mathcal{T}$ (see Figure 4).

Proof.

From Lemma 5, we know that the image of the action map is constrained to the plane $J_{1}+J_{2}+J_{3}=\sqrt{2h}$. This is bounded by $J_{i}\geq 0$ and hence the image is contained in the intersection of the plane $J_{1}+J_{2}+J_{3}=\sqrt{2h}$ with the positive quadrant. Since the maps $J_{i}$ are continuous on the reduced phase space, every point in the interior of $\mathcal{T}$ must be a point in the image of the action map. On the boundary of $\mathcal{T},$ the lines $J_{1}=0$ and $J_{3}=0$ are the image of the lines $\mathcal{L}_{1}$ and $\mathcal{L}_{4}$ respectively while $J_{2}=0$ is the image of the cyan curve $\mathcal{C}$ together with the yellow and purple segments of $\mathcal{L}_{2}$ and $\mathcal{L}_{3}$. Thus, the triangle $\mathcal{T}$ formed by intersecting the plane $J_{1}+J_{2}+J_{3}=\sqrt{2h}$ with the positive quadrant is the image of the symmetry reduced phase space under the action map. ∎

The action map calculated using (31) is shown in Figure 4 for $h=\frac{1}{2}$. Let lines $J_{i}=0$ be $\mathfrak{J}_{i}$ and call the interior lines $\gamma_{1}$ (red and grey) and $\gamma_{2}$ (magenta and orange). We denote by $A_{ij}$ the intersection of $\mathfrak{J}_{i}$ and $\gamma_{j}$. Similar polytopes are known to classify toric systems (two dimensional integrable systems where both integrals are global $S^{1}$ actions). For more details on this, see [12]. In our case, the $\gamma_{i}$ in the interior of the action map reflect the non-toric nature of the ellipsoidal integrable system. Along these lines the hyperelliptic action integrals (9) become elliptic. Let

[TABLE]

where $k^{2}=\frac{(e_{4}-e_{3})(e_{2}-e_{1})}{(e_{4}-e_{2})(e_{3}-e_{1})}$ and $K(k),\ \Pi(\alpha,k)$ are the complete elliptic integrals of the first and third kind respectively. The tangential intersections of $\mathfrak{J}_{2}$ with $\gamma_{1},\gamma_{2}$ occur at $A_{21}$ and $A_{22}$. These are given by

[TABLE]

where $\alpha_{1}=\frac{e_{2}-e_{1}}{e_{2}-e_{4}}$ and $\alpha_{3}=\frac{e_{4}-e_{3}}{e_{1}-e_{3}}$. Transverse intersections of $\mathfrak{J}_{i}$ with $\gamma_{j}$ occur at $A_{31}$ and $A_{12}$ given by

[TABLE]

where $u_{1}=\sqrt{\frac{e_{1}-e_{2}}{e_{1}-e_{3}}}$ and $u_{2}=\sqrt{\frac{e_{2}-e_{3}}{e_{2}-e_{4}}}$. The intersection of $\gamma_{1}$ and $\gamma_{2}$ is located at

[TABLE]

where $v_{1}=\sqrt{\frac{e_{1}-e_{2}}{e_{1}-e_{4}}}$ and $v_{2}=\sqrt{\frac{e_{1}-e_{3}}{e_{1}-e_{4}}}$. Notice that these intersection points are invariant under affine transformations of $(e_{1},e_{2},e_{3},e_{4})$ as expected.

The action map $(J_{1},J_{2},J_{3})$ is equivalent to the map $(J_{1},J_{1}+J_{2}+J_{3},J_{3})$ by a uni-modular transformation. Since $J_{1}+J_{2}+J_{3}=1$ for the reduced system on $S^{2}\times S^{2}$, we have 2 actions only, ie. $(J_{1},J_{3})$. By using similar transformations, any pair $(J_{i},J_{j})$ can be chosen as the actions for the reduced system. The resulting image of the corresponding action map is the projection of the action map in Figure 4 onto the $J_{i}J_{j}$ plane.

Corolary 11**.**

A possible set of actions for the reduced ellipsoidal integrable system is $(J_{1},J_{3})$. The image of the reduced space $S^{2}\times S^{2}$ under this map is a right-angled isosceles triangle obtained from projecting Figure 4 onto the $J_{2}=0$ plane.

Proof.

The reduction can be done in appropriate action variables directly. Since $J_{1}+J_{2}+J_{3}$ is a global action variable that is equal to the square root of twice the Hamiltonian, reduction means the following two things. Fix the action (i.e. fix the energy), and quotient by its flow. The flow of this action only changes it’s conjugate angle, and so the quotient identifies this angle to a point. The remaining system with two degrees of freedom has action variables $J_{1},J_{3}$, and the image of the action map of the reduced system is the projection of the “spatial” fixed energy triangle in Figure 4 a) onto the appropriate coordinate plane in Figure 4 b). ∎

The choice of which action variable to present the reduced system in is somewhat arbitrary, and we prefer not to make any choice and hence keep showing the “spatial” picture of the triangle in $J_{1}J_{2}J_{3}$-space in the following section, even when discussing the reduced system on $S^{2}\times S^{2}$. In general an integrable system can be represented by its energy surface in action space, which means the surface $H(J_{1},J_{2},J_{3})=h$, which will depend on $h$, and may not even be continuous. However, in our setting we have simplest possible case of a maximally superintegrable system for which $H(J_{1}+J_{2}+J_{3})=h$, and so the triangle in action space occurs for every superintegrable system for which globally continuous actions can be defined.

The position of the hyperbolic-hyperbolic point in the image of the action map (the intersections of the lines $\gamma_{1}$ and $\gamma_{2}$) is uniquely determined by the reduced parameters $1<a<b$ of the system. In fact, the map $(1/b,a/b)\mapsto\frac{2}{\pi}(\sin^{-1}\sqrt{1/b},\cos^{-1}\sqrt{a/b})$ maps the triangle in parameter space Figure 2 a) to the triangle in action space Figure 4 b). The involution in parameter space (22) becomes the reflection across the diagonal $(J_{1},J_{3})\mapsto(J_{3},J_{1})$.

Performing the affine transformation of $e_{i}$ directly in the action integral and applying the same transformation to the integration variable $s$ does change the integral. However, then also transforming $(\eta_{1},\eta_{2})$ according to Lemma 1 recovers the original integral, as expected.

6 Degenerate Systems on $S^{2}\times S^{2}$

In this section, we study all systems arising from separating the geodesic flow in degenerate coordinates on $S^{3}$. We begin by focusing on the following $3$ systems: prolate $(1\ (2\ 3)\ 4)$, oblate $(1\ 2\ (3\ 4))$ and Lamé $(1\ (2\ 3\ 4))$. These correspond to the edges of the Stasheff polytope. Further degenerations of these coordinates, cylindrical $((1\ 2)(3\ 4))$ and two forms of spherical $(1\ (2\ (3\ 4)),(1\ ((2\ 3)\ 4))$ form the corners of the polytope.

It will be shown that the integrable systems (reduced and un-reduced) corresponding to these degenerate coordinate systems can also be obtained by smoothly deforming their ellipsoidal counterparts. Thus, we will establish the analogue of the result by Schöbel and Veselov [38], namely that the correct moduli space for this family of integrable systems is the Stasheff polytope.

The main feature of the degenerate systems corresponding to the lower-dimensional faces of the Stasheff polytope is the appearance of global symmetries, in the case of $S^{3}$ either $SO(2)$ and $SO(3)$. The results of this section described in detail below can be sumarised in the following theorem. Consider the designation of a separable coordinate system on $S^{3}$ by pairs of nested brackets inserted into 4 objects as shown in Figure 1.

Theorem 3**.**

For each pair of brackets that enclose two adjacent members, the corresponding (reduced) integrable system has an $SO(2)$ symmetry. For each pair of brackets that enclose three adjacent members the corresponding (reduced) integrable system has a global $SO(3)$ symmetry. The generic ellipsoidal integrable system with quadratic integrals degenerates to an integrable system with quadratic integrals. If there is an $SO(k)$ symmetry the corresponding quadratic integral is replaced by its square root. For $SO(2)$ this gives a global $S^{1}$ action, while for $SO(3)$ this gives an almost global $S^{1}$ action.

In particular, the oblate, prolate, and spherical systems have one global $S^{1}$ action each, the cylindrical system has two global $S^{1}$ actions, and the spherical and the Lamé system have an almost-global $S^{1}$ action each. In addition, we find that the prolate system is generalised semi-toric and the cylindrical system is toric but the $S^{1}\times S^{1}$ action is not effective. By almost global $S^{1}$ action we mean that the action fails in the preimage of an isolated point of the image of the momentum map. The corresponding spherical singularity in the spherical system [31] also appears in the Lamé system.

6.1 Prolate Coordinates

We begin by considering prolate coordinates. It will be shown that the corresponding integrable system is generalised semi-toric and has non-trivial monodromy. In addition, the global action triangle is half of the semi-toric polygon invariant.

6.1.1 Separation of Variables

Prolate coordinates on $S^{3}$, denoted by $(1\ (2\ 3)\ 4)$, are a degeneration of ellipsoidal coordinates arising from setting the middle two semi major axes equal, i.e. $e_{2}=e_{3}$. We normalise the $e_{j}$ according to $(e_{1},e_{2}=e_{3},e_{4})=(0,1=a,b)$. From [29], an explicit representation of prolate coordinates is

[TABLE]

where $0\leq s_{1},s_{2}\leq 1\leq s_{3}\leq b$. Since $s_{2}$ is an ignorable coordinates, the Hamilton Jacobi equation can be separated easily to give integrals $(2H,G_{pro},\ell_{23})$ where $G_{pro}=b\ell_{12}^{2}+b\ell_{13}^{2}+\ell_{14}^{2}$. The corresponding momenta are

[TABLE]

where $i\in\{1,3\}$ and $(l,g)$ are the values of $\ell_{23}$ and $G_{pro}$, respectively. We call the triple $(2H,\ell_{23},G_{pro})$ on $T^{*}S^{3}$ the $1$-parameter family of prolate integrable systems. Similarly, $(\ell_{23},G_{pro})$ gives a $1$-parameter family of reduced prolate integrable systems on $S^{2}\times S^{2}$.

Note that we have chosen $\ell_{23}$ as an integral since it is naturally a global $S^{1}$ action, unlike its square. The quadratic integrals can be obtained as a limit of the ellipsoidal integrable system.

Lemma 12**.**

The integrals $\left(\ell_{23}^{2},G_{pro}\right)$ as well as the separated momenta (36) can be obtained by smoothly degenerating their ellipsoidal counterparts (13) and (14).

Proof.

The transformation from ellipsoidal to prolate coordinates is given by

[TABLE]

in the limit $\epsilon\to 0$ where $\tilde{s}_{2}\in[0,1]$. The transformation from $(s_{2},p_{2})$ to $(\tilde{s_{2}},\tilde{p_{2}})$ is canonical.

Let $(\tilde{\eta}_{1},\tilde{\eta}_{2})=\left.(\eta_{1},\eta_{2})\right|_{e_{3}=e_{2}+\epsilon}.$ Substituting (37) into (13) and taking the limit gives

[TABLE]

where we have normalised the $e_{j}$ by setting $(e_{1},e_{2}=e_{3},e_{4})=(0,1,b)$.

For the separated equations, we insert (37) into (14) and expand about $\epsilon=0$ to obtain

[TABLE]

Taking the limit of (38) as $\epsilon\to 0$ and dropping the tildes gives (36). ∎

In the prolate limit, $F_{2}$ and $F_{3}$ become singular. However, multiplying both integrals by $(e_{3}-e_{2})$ gives

[TABLE]

The other two integrals $F_{1}$ and $F_{4}$ degenerate smoothly to

[TABLE]

6.1.2 Critical Points and Momentum Map

Since the integrals of the prolate system are significantly simpler than those of the ellipsoidal system, we can easily compute the critical points and values directly. However, it is interesting to note that we can also use the method of compatible Poisson structures with the matrix $C=\text{diag}(0,1,1+\epsilon,b)$ for $0<\epsilon<b-1$ for this computation.

Using Proposition 3 with this $C$, we get

[TABLE]

with $(I_{1},I_{2})=(G_{pro}+2h+(b-1)\ell_{23}^{2}+\epsilon(\ell_{12}^{2}+\ell_{14}^{2}+\ell_{24}^{2}),G_{pro}+\epsilon(b\ell_{12}^{2}+\ell_{14}^{2}))$. While the integrals $I_{1}$ and $I_{2}$ may appear complicated, the system $(I_{1},I_{2})$ is equivalent to the system $(G_{pro},\ell_{23}^{2})$ in the limit as $\epsilon\to 0$. We can find the critical points and values of the system $(I_{1},I_{2})$ then taking the limit as $\epsilon\to 0$ at the end of calculations to recover the correct results for the system $(G_{pro},\ell_{23}^{2})$.

Proposition 13**.**

The momentum map for the reduced prolate integrable system is the region bounded by the curve $\mathcal{P}_{1}:G_{pro}=b(1-\ell_{23}^{2})$ and line $\mathcal{P}_{2}:G_{pro}=0$ shown in Figure 5 a). There is an isolated critical value at $(0,1)$.

Proof.

Critical points of the system $(\ell_{23}^{2},G_{pro})$ can be computed directly or by applying Proposition 4:

The blue line $\mathcal{P}_{2}:G_{pro}=0$ has $\lambda=0$ and the critical points are parametrised by $\bm{L}=(0,0,0,\ell_{23},\ell_{24},\ell_{34})$ with $\ell_{23}^{2}+\ell_{24}^{2}+\ell_{34}^{3}=1$ and $\ell_{23}=l$. These are co-dimension 1 elliptic points. 2. 2.

The green line $\mathcal{P}_{1}:G_{pro}=b(1-\ell_{23}^{2})$ in Figure 5 b) has $\lambda=a$ and the critical points are parametrised by $\bm{L}=(\ell_{12},\ell_{13},0,\ell_{23},0,0)$ with $\ell_{12}^{2}+\ell_{13}^{2}+\ell_{23}^{3}=1$ and $\ell_{23}=l$. These are co-dimension 1 elliptic point. 3. 3.

The lines $\ell_{23}^{2}=0$ has both $\lambda=1$, $\lambda=1+\epsilon$ as well as the curve $I_{2}=\frac{I_{1}^{2}}{8h}$ corresponding to the double root $\lambda\in[1,1+\epsilon]$. These are degenerate critical values of system $(\ell_{23}^{2},G_{pro})$.

For the reduced prolate integrable system $(\ell_{23},G_{pro})$, we see that the line $\mathcal{P}_{2}$ remains critical and has $l\in[-1,1]$. The line $P_{1}$ for $(\ell_{23}^{2},G_{pro})$ becomes a parabola for $(\ell_{23},G_{pro})$. The line $\ell_{23}=0$ becomes regular values after changing from $\ell_{23}^{2}$ to $\ell_{23}$ with the exception of the isolated point $(\ell_{23},G_{pro})=(0,1)$ which has critical points $\ell_{14}=\pm 1$. This is a focus-focus point and its fibre on $S^{2}\times S^{2}$ a doubly pinched torus. ∎

6.1.3 Action Map and Monodromy

From (36) and the same reasoning used to obtain (31), we have the following formulae for the actions

[TABLE]

where the $p_{k}$ are given in (36). Here $(r_{1},r_{2})$ are the roots of $p_{1,3}^{2}$ from (36) where $0\leq r_{1}\leq 1\leq r_{2}\leq b$. Note that $J_{2}$ simplifies to $\left|\ell_{23}\right|$. Like for the ellipsoidal system, the prolate actions also satisfy (32). The action map for the prolate system is shown in Figure 6 a) where the black dot corresponding to the focus-focus point is located at $\frac{2}{\pi}(\sin^{-1}(\frac{1}{b}),0,\frac{\pi}{2}-\sin^{-1}(\frac{1}{b}))$.

The reduced prolate system is a two degree of freedom integrable system where one of the integrals is a global $S^{1}$ action and all singularities are either elliptic or focus-focus type. Thus, we have the following.

Corolary 14**.**

The reduced prolate system $(\ell_{23},G_{pro})$ on $S^{2}\times S^{2}$ is a generalised semi-toric system.

Semi-toric systems have been globally classified using $5$ symplectic invariants [37]. One of these is the polygon invariant, which is a family of rational convex polygons. This is a generalisation of the Delzant polytope (see, e.g., [12]) and allows us to compare the standard affine structure of $\mathbb{R}^{2}$ with that of the momentum map [2, 41]. In Figure 6 b) we show one representative of the polygon invariant. This is simply the projection of the action map onto the $(\ell_{23},J_{1})$ axes with both signs of $\ell_{23}$ considered. The red vertex at $(0,1)$ is a fake corner and is the result of “opening up” from $|\ell_{23}|$ to $\ell_{23}$. For more information on the classification of semi-toric systems, see [41, 2].

Another symplectic invariant of a semi-toric system is the height invariant, which is the position of the focus-focus point in the image of the action map. Note that this is the limit of the image of the hyperbolic-hyperbolic point in the action map for the degeneration $a=1$. Specialising (35) to this case gives $\frac{2}{\pi}\cos^{-1}\sqrt{1/b}$ for the height invariant.

Another property of semi-toric systems is the non-trivial monodromy of the actions; the focus-focus equilibrium implies that one can only locally construct a smooth set of action variables. Monodromy has been well studied, both classically and quantum mechanically, see e.g. [11, 16, 10, 8]. The prolate system has non-trivial monodromy.

Lemma 15**.**

The reduced prolate system has non-trivial monodromy with monodromy matrix

[TABLE]

Proof.

Let $C_{1}$ and $C_{3}$ be cycles that enclose the intervals $[0,\min(1,r_{1})]$ and $[\max(1,r_{2}),b]$ respectively. We rewrite (39) as

[TABLE]

We observe that if $(l,g)=(0,g>1)$ then $r_{1}\to 1^{-}$. while if $g_{pro}<1$, then $r_{2}\to 1^{+}$. We now show that the actions (41), while continuous everywhere, are not globally smooth. Consider the slope of the action $J_{i}$ considered as a function of $l$,

[TABLE]

When $l=0$, if $C_{i}$ encloses $s_{i}=1$ then $W_{i}=-\text{sgn}(l)$, otherwise it vanishes. From our analysis of $C_{1}$ and $C_{2}$ around $s_{i}=1$ we have

[TABLE]

where $(\kappa_{1},\kappa_{3})=(1,0)$ when $g_{pro}<1$, otherwise $(\kappa_{1},\kappa_{3})=(0,1)$. Thus, the actions $J_{1}$ and $J_{3}$ are continuous but not differentiable at $l=0$.

For $l>0$, let $J_{+}=\left(J_{1},J_{2},J_{3}\right)^{t}$ and similarly for $J_{-}$. Note that $J_{1}$ and $J_{3}$ are even functions of $l$ while $J_{2}$ is odd. This means $J_{-}(-l)=SJ_{+}(l)$ where $S=\text{diag}(1,-1,1)$. We are now interested in finding unimodular matricies $M_{1},M_{2}\in SL(3,\mathbb{Z})$ such that $J_{+}$ and $M_{i}J_{-}$ are locally smooth across $l=0$. To ensure continuity at $l=0$ we require

[TABLE]

The above relations imply that $\left(J_{1},0,J_{3}\right)^{t}$ is an eigenvector of both $M_{1}$ and $M_{2}$ with eigenvalue $+1$. For arbitrary $J_{1},J_{3}$ the corresponding eigenvector equation implies that $M_{i}$ has the form

[TABLE]

For the actions to be smoothly joined when $g>1$ we require

[TABLE]

and similarly for $g<1$ and $M_{2}$. The limits in (42) force $(\alpha_{1},\beta_{1})=(0,-2)$ and $(\alpha_{2},\beta_{2})=(-2,0)$. The corresponding monodromy matrix is given by $M=(M_{2}S)^{-1}(M_{1}S)$ which we compute to be (40). ∎

The monodromy of the reduce system with actions $(J_{1},\ell_{23})$ is obtained by the restriction to the top left block of $\mathfrak{M}$. It should be stressed that the integrable system with Hamiltonian $H$ does not have monodromy, it is superintegrable, and does not even have dynamically defined tori. However, the fibration defined by the three commuting functions $(H,\ell_{23},G_{pro})$ on $T^{*}S^{3}$ has monodromy as computed. Similarly, the commuting function $(\ell_{23},G_{pro})$ on $S^{2}\times S^{2}$ have monodromy given by the top left block of $\mathfrak{M}$.

6.2 Oblate Coordinates

Eventhough the definition of these coordinates is similar to the prolate case, the corresponding integrable system is significantly different. In particular, even though it does have a global $S^{1}$ action it is not semi-toric because of the appearance of hyperbolic and degenerate singularities.

6.2.1 Separation of Variables

Oblate coordinates, denoted by $(1\ 2\ (3\ 4))$ and $((1\ 2)\ 3\ 4)$, lie on opposite sides of the dotted line in Figure 1. They are equivalent by flipping the ordering of the $e_{i}$ by applying $e_{i}\mapsto-e_{i}$ then reordering. Consequently, the corresponding integrable systems are equivalent. We focus on the $(1\ 2\ (3\ 4))$ coordinates and normalise according to $(e_{1},e_{2},e_{3}=e_{4})=(0,1,a)$. This system is equivalent to the $((1\ 2)\ 3\ 4)$ with $(e_{1}=e_{2},e_{3},e_{4})=(0,1,\frac{a}{a-1})$. An explicit definition of these coordinates is given by

[TABLE]

where $0\leq s_{1},s_{3}\leq 1\leq s_{2}\leq a$. The integrals are $(2H,\ell_{34},G_{obl})$ where $G_{obl}=a\ell_{12}^{2}+\ell_{13}^{2}+\ell_{14}^{2}$. Let $(l,g)$ be functional values of $\ell_{34}$ and $G_{obl}$ respectively, we obtain the separated momenta

[TABLE]

where $i\in\{1,2\}$. We call the triple $(2H,\ell_{34},G_{obl})$ on $T^{*}S^{3}$ the 1-parameter family of oblate integrable systems and $(\ell_{34},G_{obl})$ the corresponding reduced oblate integrable system.

We have a similar result to Lemma 12 for the oblate system.

Lemma 16**.**

The integrals $(\ell_{34}^{2},G_{obl})$ as well as the separated momenta (44) can be obtained by smoothly degenerating their ellipsoidal counterparts (13) and (14).

Proof.

Using the transformation

[TABLE]

where $\tilde{s}_{3}\in[0,1]$ and following the same procedure as Lemma 12 gives the result. ∎

In the oblate limit, the Uhlenbeck integrals $\tilde{F}_{i}$ are

[TABLE]

6.2.2 Critical Points and Momentum Map

We can also use the method of compatible Poisson structures to aid in studying this system. Using the matrix $C=\text{\text{diag}}(0,1,a,a+\epsilon)$ in Proposition 3 gives the equation

[TABLE]

where $(I_{1},I_{2})=(G_{obl}+2ah-(a-1)\ell_{34}^{2}+\epsilon(\ell_{12}^{2}+\ell_{13}^{2}+\ell_{23}^{2}),-aG_{obl}-a\epsilon\ell_{12}^{2}-\epsilon\ell_{13}^{2})$. According to Corollary 5, the critical values occurs at the curve $I_{2}=\frac{I_{1}^{2}}{8h}$ as well as the lines $I_{2}=0$, $I_{2}=2h-I_{1}$, $I_{2}=2ha^{2}-aI_{1}$ and $I_{2}=2h(a+\epsilon)^{2}-(a+\epsilon)I_{1}$ when $\lambda=0,1,a,a+\epsilon$ respectively.

Proposition 17**.**

The critical values of the momentum map for the reduced oblate integrable system are the curves $\mathcal{O}_{1}:G_{obl}=(\sqrt{a}-\sqrt{a-1}\left|\ell_{34}\right|)^{2},\mathcal{O}_{2}:G_{obl}=1-\ell_{34}^{2}$ and $\mathcal{O}_{3}:G_{obl}=0$. The momentum map is shown in Figure 7 a) with $2h=1$.

Proof.

Applying Proposition 4 and following a similar calculation as the ellipsoidal case we get:

The blue line $\mathcal{O}_{3}:G_{obl}=0$ has $\lambda=0$ and critical points parametrised by $\bm{L}=(0,0,0,\ell_{23},\ell_{24},\ell_{34})$ with $\ell_{23}^{2}+\ell_{24}^{2}+\ell_{34}^{3}=1$ and $\ell_{34}=l_{obl}$. 2. 2.

The red and yellow curve $\mathcal{O}_{2}:G_{obl}=1-\ell_{34}^{2}$ has $\lambda=1$ and the critical points are parametrised by $\bm{L}=(0,\ell_{13},\ell_{14},0,0,\ell_{34})$ with $\ell_{13}^{2}+\ell_{14}^{2}+\ell_{34}^{3}=1$ and $\ell_{34}=l_{obl}$. 3. 3.

The cyan curve $\mathcal{O}_{1}:G_{obl}=(\sqrt{a}-\sqrt{a-1}\left|\ell_{34}\right|)^{2}$ has $\lambda\in[1,a]$ and critical points

[TABLE]

with Plücker relation $a_{1}^{2}=b_{2}^{2}+b_{3}^{2}$ and $\bm{L}\cdot\bm{L}=\frac{1}{2}a(a-1)(b_{2}^{2}+b_{3}^{2})=1$. This forces $a_{1}=\pm\frac{\sqrt{2}}{\sqrt{a(a-1)}}$ and gives a parametrisation $(\ell_{34},G_{obl})=\left(\mp\frac{(a-\lambda)}{\sqrt{a(a-1)}},\frac{\lambda^{2}}{a}\right)$ in terms of $\lambda$. This curve exists only for $0<|\ell_{34}|<\sqrt{\frac{a-1}{a}}$ which corresponds to a double root $1<\lambda<a$.

The lines $I_{2}=2ha^{2}-aI_{1}$ and $I_{2}=2h(a+\epsilon)^{2}-(a+\epsilon)I_{1}$ both gives the line $\ell_{34}=0$ (orange and magenta segments in Figure 7 b)) which are degenerate critical values for the system $(\ell_{34}^{2},G_{obl})$ since the vector field generated by $\ell_{34}^{2}$ vanishes at $\ell_{34}=0$. However, for the oblate integrable system $(\ell_{34},G_{obl})$ the line $\ell_{34}=0$ becomes a set of regular values (except at three points). Direct computation using the vector fields of $(\ell_{34},G_{obl})$ confirms these results. ∎

Let the intersection of $\mathcal{O}_{i}$ and $\mathcal{O}_{j}$ be denoted by $o_{ij\pm}$ where the sign is determined by whether the intersection occurs for a positive or negative value of $\ell_{34}$. The intersections $o_{12\pm}$ at $(\ell_{34},G_{obl})=(\pm\sqrt{\frac{a-1}{a}},\frac{1}{a})$ are tangential. The other $3$ intersections at $o_{11}=(0,a)$ and $o_{23\pm}=(\pm 1,0)$ are transverse.

The tangential intersections $o_{12\pm}$ are degenerate pitchfork singularities and their fibres are single circles $S^{1}$ on $S^{2}\times S^{2}$. The points $o_{23\pm}$ are of elliptic-elliptic type. The point $o_{11}$ is also elliptic-elliptic with 2 critical points $\bm{L}=(\pm 1,0,0,0,0,0)$.

The curves $\mathcal{O}_{1},\mathcal{O}_{3}$ as well as the yellow parts of $\mathcal{O}_{2}$ are all codimension one elliptic. The fibre of $\mathcal{O}_{3}$ and the yellow segments are single $S^{1}$, while the fibre of $\mathcal{O}_{1}$ is $2S^{1}$. The red part of $\mathcal{O}_{2}$ is codimension one hyperbolic and its fibre is $B\times S^{1}$.

The fibre of a regular value in chamber $I$ is $T^{2}$ while the fibre of a regular value in chamber $II$ is $2T^{2}$.

6.2.3 Actions

Like in the prolate case, one action for the oblate system $J_{3,obl}\coloneqq\left|\ell_{34}\right|$ is trivial. The other two non trivial actions are

[TABLE]

where $r_{2}\geq 1$ and $0\leq r_{1}\leq r_{2}\leq a$. Theorem 2 also applies here. The action map is shown in Figure 8. For the interior (red) curve $\gamma_{obl}$ we have $r_{1}=1$, $r_{2}=a(1-l^{2})\geq 1$ with $g=1-l^{2}$ for $|l|\leq\sqrt{\frac{a-1}{a}}$. The parametrisation of this curve in terms of the angular momentum $l$ where $|l|\leq\sqrt{\frac{a-1}{a}}$ is given by

[TABLE]

where $t_{1}=\frac{1}{\sqrt{a(1-l^{2})}}$ and $t_{2}=\frac{\sqrt{a(1-l^{2})-1}}{l}$. Call the intersections of $\gamma_{obl}$ with the boundary of the action map $\mathfrak{O}_{1}$ (yellow/cyan) and $\mathfrak{O}_{2}$ (magenta/orange). The point $\mathfrak{O}_{1}$ has coordinates $\gamma_{obl}(\sqrt{\frac{a-1}{a}})=(1-\sqrt{\frac{a-1}{a}},0,\sqrt{\frac{a-1}{a}})$, while $\mathfrak{O}_{2}$ is located at $\gamma_{obl}(0)=\frac{2}{\pi}(\sin^{-1}\left(\frac{1}{\sqrt{a}}\right),\cos^{-1}\left(\frac{1}{\sqrt{a}}\right),0)$.

6.3 Lamé Coordinates

The Lamé system is unusual in a number of ways. In this case we actually need to make use of the fact that the parameters $e_{i}$ live on the real projective line. This is the reason why this case is not visible in the original normalised $ab$-parameter space and a blow-up is required. Furthermore, it is a family that has larger symmetry group $SO(3)$. A larger symmetry group is in some sense related to super-integrability, however, since we don’t have a Hamiltonian in the reduced system it is harder to define what this means. As we will see after another reduction, this system becomes the Euler top (see the Appendix).

6.3.1 Separation of Variables and Stäckel System

Lamé coordinates are an extension of ellipsoidal coordinates from $S^{2}$ onto $S^{3}$ and arise from limiting to three equal semi-major axes. As with oblate, there are two cases to consider: $(1\ (2\ 3\ 4))$ and $((1\ 2\ 3)\ 4)$ which are equivalent. Here we only discuss the $(1\ (2\ 3\ 4))$ coordinates which are defined as follows

[TABLE]

where $0\leq s_{1}\leq 1$ and $0\leq f_{1}\leq s_{2}\leq f_{2}\leq s_{3}\leq f_{3}$. A possible Stäckel matrix for these coordinates is

[TABLE]

with integrals $(2H,2H-F_{L},G_{L}-f_{1}(2H-F_{L}))$ where $(F_{L},G_{L})=(\ell_{12}^{2}+\ell_{13}^{2}+\ell_{14}^{2},f_{1}l_{34}^{2}+f_{2}l_{24}^{2}+f_{3}l_{23}^{2})$. From (46), the separated momenta are given by

[TABLE]

where $k=2,3$ and $(f_{L},g_{L})$ are functional values of $(F_{L},G_{L})$. We call the triple $(2H,F_{L},G_{L})$ on $T^{*}S^{3}$ the Lamé integrable system and $(F_{L},G_{L})$ the corresponding reduced Lamé integrable system on $S^{2}\times S^{2}$.

The important feature of this case is the appearance of the integral $F_{L}$ with a $SO(3)$ symmetry given by the rotations generated by $\ell_{1i}$ using $B_{L}$. Similar to the prolate and oblate families, the Lamé system can also be obtained as a limit of the ellipsoidal system.

Lemma 18**.**

The integrals $(F_{L},G_{L})$ and separated momenta (47) for the Lamé integrable system can be obtained from their ellipsoidal counterparts (13) and (14).

Proof.

A possible limiting process from ellipsoidal to Lamé coordinates is given in [28]. However in this case, it is much simpler to use the transformation

[TABLE]

for $\epsilon>0$ and $\tilde{s}_{1}\in[0,1]$. Applying (48) and taking the limit as $\epsilon\to 0$ immediately gives

[TABLE]

The separated momenta are obtained using the same method as in the prolate case. ∎

In the Lamé limit, the Uhlenbeck integrals $\tilde{F}_{i}$ are as follows

[TABLE]

Note that up to projective transformations, the system only has one parameter $\frac{f_{3}-f_{1}}{f_{2}-f_{1}}$, but we use $f_{i}$ to keep the higher symmetry.

The vector field of $G_{L}$ given by $B_{\bm{L}}\nabla G_{L}$ has a semi-direct product stucture: the equations for $\ell_{23},\ell_{24},\ell_{34}$ decouple from the others, and they are in fact Euler’s equations for the rigid body on $SO(3)$ with moments of inertia given by $f_{i}^{-1}$. The equation for the remaining three variables are linear equations with time-varying coefficients given by the solution of Euler’s equation.

6.3.2 Critical Points and Momentum Map

Using the matrix $C=\text{diag}(-\frac{1}{\epsilon},f_{1},f_{2},f_{3})$ gives the integrals

[TABLE]

We need consider the $\frac{1}{\epsilon}$ order term since the limit as $\epsilon\to 0$ of $(I_{1},I_{2})$ is infinite.

Proposition 19**.**

The set of critical values of the momentum map for the Lamé system is composed of four straight lines $\mathfrak{L}_{j}:F_{L}=1-\frac{1}{f_{j}}G_{L}$ for $j=1,2,3$ and $\mathfrak{L}_{4}:F_{L}=0$. This is shown in Figure 9 a).

Proof.

The critical points for the Lamé integrable systems are similar to those for the lines in the ellipsoidal system:

The line $\mathfrak{L}_{4}:F_{L}=0$ is the limit as $\epsilon\to 0$ of the line $I_{2}=-\frac{1}{\epsilon}(I_{1}+\frac{1}{\epsilon})$ with $\lambda=-\frac{1}{\epsilon}$ and the critical points are parametrised by $\bm{L}=(0,0,0,\ell_{23},\ell_{24},\ell_{34})$ with $\ell_{23}^{2}+\ell_{24}^{2}+\ell_{34}^{2}=1$ and $f_{1}\ell_{34}^{2}+f_{2}\ell_{24}^{2}+f_{3}\ell_{23}^{2}=g_{L}$. 2. 2.

The line $\mathfrak{L}_{1}:F_{L}=1-\frac{1}{f_{1}}G_{L}$ is the limit as $\epsilon\to 0$ of the line $I_{2}=f_{1}(I_{1}-f_{1})$ with $\lambda=f_{1}$ and the critical points are parametrised by $\bm{L}=(0,\ell_{13},\ell_{14},0,0,\ell_{34})$ with $\ell_{13}^{2}+\ell_{14}^{2}+\ell_{34}^{2}=1$ and $f_{1}\ell_{34}^{2}=g_{L}$. 3. 3.

The line $\mathfrak{L}_{2}:F_{L}=1-\frac{1}{f_{2}}G_{L}$ is the limit as $\epsilon\to 0$ of the line $I_{2}=f_{2}(I_{1}-f_{2})$ with $\lambda=f_{2}$ and the critical points are parametrised by $\bm{L}=(\ell_{12},0,\ell_{14},0,\ell_{24},0)$ with $\ell_{12}^{2}+\ell_{14}^{2}+\ell_{24}^{2}=1$ and $f_{2}\ell_{24}^{2}=g_{L}$. 4. 4.

The line $\mathfrak{L}_{3}:F_{L}=1-\frac{1}{f_{3}}G_{L}$ is the limit as $\epsilon\to 0$ of the line $I_{2}=f_{3}(I_{1}-f_{3})$ with $\lambda=f_{3}$ and the critical points are parametrised by $\bm{L}=(\ell_{12},\ell_{13},0,\ell_{23},0,0)$ with $\ell_{12}^{2}+\ell_{13}^{2}+\ell_{23}^{2}=1$ and $f_{3}\ell_{23}^{2}=g_{L}$.

The curve $I_{2}=\frac{I_{1}^{2}}{8h}$ for $\lambda\in[f_{1},f_{2}]$ shrinks to the degenerate point $(F_{L},G_{L})=(1,0)$ in the limit $\epsilon\to 0$. ∎

Let the intersections of $\mathfrak{L}_{i}$ and $\mathfrak{L}_{j}$ be denoted by $t_{ij}$ and the three way intersection of $\mathfrak{L}_{1}$, $\mathfrak{L}_{2}$, $\mathfrak{L}_{3}$ by $T_{123}$. The intersections $t_{14}$ and $t_{34}$ are elliptic-elliptic critical values with 2 points in their fibres $2S^{1}$ on $S^{2}\times S^{2}$. The point $t_{24}$ is elliptic hyperbolic and its fibre is $C_{2}$.

Lemma 20**.**

The three-way intersection $T_{123}$ at $(F_{L},G_{L})=(1,0)$ is a degenerate singularity of spherical type.

Proof.

The linearisation of the vector field generated by $F_{L}$ and $G_{L}$ is $\nabla_{\bm{L}}B_{\bm{L}}(\alpha\nabla_{\bm{L}}F_{L}+\beta\nabla_{\bm{L}}G_{L})$. At the three-way intersection $T_{123}$, this matrix becomes

[TABLE]

where $\bm{0}$ is a $3\times 3$ matrix of all zeros. The eigenvalue of (49) is $(0,0,0,0,0,0)$ meaning that $T_{123}$ is a degenerate critical value. The rank of the differential of the moment map drops by $1$ at $T_{123}$. This is known as a spherical type singularity studied in the thesis [31]. Systems with a spherical type singularity are characterised by the presence of a globally defined, continuous but not smooth action. The fibre of a spherical type singularity is diffeomorphic to a product of spheres. In this case the preimage of $T_{123}$ is $S^{2}$ and every point in this fibre is critical. ∎

It was shown in [31] that the geodesic flow on $S^{n}$ in polyspherical coordinates gives rise to systems containing a spherical type singularity. While the Lamé coordinate system is not polyspherical, the reduced Lamé integrable system is an example of a system that has a spherical singularity that was not studied in [31]. Since the Lamé coordinate system is obtained by extending the ellipsoidal coordinates on $S^{2}$ to $S^{3}$, we can conjecture that systems originating from the geodesic flow on $S^{n}$ in coordinates obtained from extending a coordinate system from $S^{k}$ to $S^{n}$ where $2\leq k<n$ will contain a spherical type singularity.

6.3.3 Actions

The actions of the Lamé system are given by

[TABLE]

where $r_{2}=\frac{g_{L}}{1-f_{L}}$. Theorem 2 applies for the Lamé system also. The first action evaluates to

[TABLE]

Notice that $F_{L}$ has a vector field $B_{\bm{L}}\nabla F_{L}$ has a flow that is the rotation of $(\ell_{12},\ell_{13},\ell_{14})$ about the fixed axis given by $(\ell_{34},-\ell_{24},\ell_{23})$. The frequency of this rotation is given by the length of the axis, and is hence not constant.

Lemma 21**.**

$J_{1}=1-\sqrt{1-F_{L}}$* is an almost global $S^{1}$-action.*

Proof.

The vector field $B_{\bm{L}}\nabla J_{1}$ has periodic flow which is given by the rotation about the same axis as the flow of $F_{L}$, but here the axis is normalised because we need to divide by $\sqrt{1-F_{L}}$ and using the Casimir $2h=1$ this means to divide by the length of the axis. It is only “almost” global because when $2h-F_{L}=\ell_{34}^{2}+\ell_{24}^{2}+\ell_{23}^{2}=0$ the normalisation factor vanishes and the vector field is not defined. Because of $2h=1$ this occurs on the sphere $S^{2}$ given by $\ell_{12}^{2}+\ell_{13}^{2}+\ell_{14}^{2}=1$. ∎

Note that (50) means that lines of constant $f_{L}$ correspond to lines of constant $J_{1}$ in action space. In general, we have

Lemma 22**.**

Straight lines in the image of the momentum map $(F_{L},G_{L})$ of the reduced Lamé system on $S^{2}\times S^{2}$ given by $F_{L}=1-\frac{1}{r_{2}}G_{L}$ maps to straight lines in action space.

Proof.

Observe that

[TABLE]

where $\gamma=\min(r_{2},f_{2})$, $r_{2}=\frac{g_{L}}{1-f_{L}}$ and $\mathcal{F}$ is function of $r_{2}$ only. Since $J_{3,L}=1-J_{1,L}-J_{2}$, this implies that the image of a straight line $F_{L}=1-\frac{1}{r_{2}}G_{L}$ with constant slope $\frac{1}{r_{2}}$ under the action map is again a straight line. ∎

Figure 9 b) shows an example of the action map. Let the magenta line in the interior of the action map be denoted by $\mathfrak{L}_{M}$. This has parameterisation

[TABLE]

where $\Delta=\sqrt{\frac{f_{1}-f_{2}}{f_{1}-f_{3}}}$. The line intersects the boundary of the action map at $(1,0,0)$ and $\frac{2}{\pi}(0,\sin^{-1}(\Delta),\cos^{-1}(\Delta)$.

The fact that lines of constant $f_{L}$ and lines through $T_{123}$ are mapped to straight lines does not imply it is a linear map, because the map along these lines is determined by the non-linear map $\mathcal{F}$.

Since away from $f_{l}=1$, the action variable $J_{1}$ is defined we can consider reduction with respect to the flow of $J_{1}$ on levels with $0<f_{L}<1$. Fixing the action and identifying the corresponding angle variable to a point gives the action $J_{2}$ for that constant value of $f_{L}$, and up to an overall constant factor this is the action of the Euler top.

6.4 Spherical Coordinates

Spherical coordinates (or rather poly-spherical coordinates) correspond to the case where simultaneously there is an $SO(2)$ and an $SO(3)$ symmetry. Accordingly we do have a global $S^{1}$ action. However, the induced integrable system on $S^{2}\times S^{2}$ is not semi-toric because it has a degenerate point, which corresponds to the critical values at which the almost global action is not differentiable.

6.4.1 Separation of Variables

The two forms of spherical coordinates are found by setting $f_{1}=f_{2}$ or $f_{2}=f_{3}$ in Lamé coordinates. We call these the $12$ and $23$ spherical coordinates. The two systems are equivalent by a permutation of coordinates. These can also be obtained by setting $a=1$ in prolate and oblate coordinates, respectively. The $23-$spherical coordinate system $(1\ (2\ (3\ 4)))$ is defined by

[TABLE]

where $0\leq s_{k}\leq 1$ and $k=1,2,3$. Due to the simplicity of these coordinates, we can manually separate the corresponding Hamilton-Jacobi equation. The geodesic Hamiltonian can be expressed as

[TABLE]

The integrals are $(2H_{23},\ell_{34},G_{23})$ with separated momenta

[TABLE]

where $G_{23}=\ell_{12}^{2}+\ell_{13}^{2}+\ell_{14}^{2}$ and $(l_{34},g_{23})$ are functional values of $(\ell_{34}^{2},G_{23})$.

To obtain $(\ell_{34},G_{23})$ and (53) from the Lamé system, we set $(f_{3},s_{3},p_{3})=(f_{2}+\epsilon,f_{2}+\epsilon\tilde{s}_{3},\tilde{p}_{3}/\epsilon)$ where $\tilde{s}_{3}\in[0,1]$ and normalise $(f_{1},f_{2})=(0,1)$. To come from oblate, we let $(a,s_{3},p_{3})=(1+\epsilon,1+\epsilon\tilde{s}_{3},\tilde{p}_{3}/\epsilon)$.

6.4.2 Critical Points and Momentum Map

The critical points and values are easily obtained by direct computation to give

Proposition 23**.**

The image of momentum map for the $23-$spherical system $(\ell_{34},G_{23})$ with $2h=1$ has critical values $\mathfrak{C}_{1}:G_{23}=1-\ell_{34}^{2}$ and $\mathfrak{C}_{2}:G_{23}=0$ which are both codimension one elliptic (see Figure 10).

Proof.

The computation of critical points and values are straight forward for this system. ∎

The fibre of a regular value on $S^{2}\times S^{2}$ is a torus $T^{2}$ with multiplicity one. The fibres along $\mathfrak{C}_{1}$ and $\mathfrak{C}_{2}$ are single $S^{1}$. The intersections of $\mathfrak{C}_{1}$ and $\mathfrak{C}_{2}$ are codimension 2 elliptic points and have 1 critical point in their fibres. The linearisation $\nabla_{\bm{L}}B_{\bm{L}}(\alpha\nabla_{\bm{L}}G_{23}+\beta\nabla_{\bm{L}}\ell_{34})$ has eigenvalues $(0,0,-i\beta,i\beta,-i\beta,i\beta)$ at $D_{23}=(0,1)$ making the peak of the parabola $D_{23}$ a degenerate singularity. Similar to the Lamé system the rank of the differential of the moment map drops by 1 at $D_{23}$ and it’s fibre is $S^{2}$. This is also an example of a spherical type singularity. Spherical coordinates is a type of polyspherical coordinates and these have been studied in detail in [31].

In the limit $a\to 1$, the bifurcation diagram for the oblate coordinates in Figure 7 a) degenerates to Figure 10 a). In particular, the elliptic-elliptic point $o_{11}$ collides with the hyperbolic line $\mathcal{O}_{2}$ and becomes degenerate. Similarly, setting $f_{2}=f_{3}$ in the Lamé system causes the elliptic-hyperbolic point $t_{24}$ to collide with the elliptic-elliptic point at $t_{34}$ while $T_{123}$ remains degenerate.

6.4.3 Actions

The action variables for the $23-$spherical system are given by

[TABLE]

where $0\leq r_{1}=g_{23}\leq 1$ and $0\leq r_{2}=1-\frac{l_{23}^{2}}{1-g_{23}}\leq 1$. We can simplify the non trivial actions to

[TABLE]

The action map is shown in Figure 10 b). Note that $(\ell_{34},\sqrt{1-G_{23}})$ defines continuous global action variables that are not differentiable at $G_{23}=1$. This is a system obtained from toric degeneration, see [31].

6.5 Cylindrical Coordinates

The coordinate system with the highest symmetry has two global $S^{1}$ action, and there is only a single point in the Stasheff polytope for which this happens. The corresponding reduced system on $S^{2}\times S^{2}$ is toric.

6.5.1 Separation of Variables

The cylindrical coordinates (also called Hopf coordinates) $((1\ 2)\ (3\ 4))$ are a further degeneration of the oblate coordinates obtained by setting both $e_{1}=e_{2}$ and $e_{3}=e_{4}$. Specifically, the transformation $(e_{2},e_{4})\to(e_{1}+\epsilon,e_{3}+\epsilon)$ along with $(s_{1},p_{1},s_{3},p_{3})\to(e_{1}+\epsilon\tilde{s}_{1},\tilde{p_{1}}/\epsilon,e_{3}+\epsilon\tilde{s}_{3},\tilde{p_{3}}/\epsilon)$ gives the following relationship between Cartesian coordinates and cylindrical coordinates:

[TABLE]

where $0\leq s_{k}\leq 1$ and $k=1,2,3$. The geodesic Hamiltonian in these coordinates is

[TABLE]

which trivially separates to give integrals $(H_{Cyl},\ell_{34}^{2},\ell_{12}^{2})$ The separated equations are

[TABLE]

where $\nu=12$ if $k=1$, $\nu=34$ if $k=3$ and $l_{\nu}$ denoted the functional value of $\ell_{\nu}$.

6.5.2 Critical Points and Momentum map

Proposition 24**.**

The bifurcation diagram for the cylindrical system $(\ell_{12},\ell_{34})$ on $S^{2}\times S^{2}$ with $2h=1$ is composed of $4$ straight lines $\ell_{34}=\pm(1\pm\ell_{12})$ which intersect transversally at $(\pm 1,0)$ and $(0,\pm 1)$ (see Figure 11).

The fibre of a regular point on $S^{2}\times S^{2}$ is $T^{2}$ with multiplicity one. The lines are all codimension one elliptic and their fibres are single $S^{1}$. The intersections of the lines are elliptic-elliptic critical values with a single critical point in their fibres.

6.5.3 Actions

The trivial actions for the cylindrical system are $(J_{1},J_{3})=(\left|\ell_{12}\right|,\left|\ell_{34}\right|)$ while the “non trivial” action is easily determined by $J_{1}+J_{2}+J_{3}=1$. The action map is shown in Figure 11 b). The relation between the symmetry reduced actions $|\ell_{12}|$ and $|\ell_{34}|$ and the global $S^{1}$ actions is to forget the absolute value sign. In this way 4 copies of the right triangle in $J_{1},J_{3}$ are glued together to a diamond in $\ell_{12},\ell_{34}$.

Proposition 25**.**

The system $(X_{1},Y_{1})$ is a toric system on $S^{2}\times S^{2}$ where $X_{1}$ and $Y_{1}$ are defined in (20).

Proof.

Both $\ell_{12}$ and $\ell_{34}$ define smooth global $S^{1}$ action on $S^{2}\times S^{2}$. However, the torus action $(\ell_{12},\ell_{34})$ is not effective. Note that $\ell_{ij}$ is the generator of rotation in the $x_{i}x_{j}$-plane represented by $\exp({it_{ij}\hat{\bm{x}}_{i}\wedge\hat{\bm{x}}_{j}})$ where $\hat{\bm{x}_{i}}$ is the unit vector in the $x_{i}$ axis. The action on the momenta $\bm{y}$ is the same. A rotation by $t_{ij}=\pi$ in the $x_{i}x_{j}$-plane has the effect

[TABLE]

on $T^{*}S^{3}$. This induces the map

[TABLE]

for $i,j,m,n$ all distinct, and

[TABLE]

for all $k\neq i,j$. In particular, the action with $t_{12}=\pi$ and $t_{34}=\pi$ both generate the same map

[TABLE]

on $S^{2}\times S^{2}$. Since the flows of $\ell_{12}$ and $\ell_{34}$ commute and (55) is an involution, we see that $(t_{12},t_{34})=(\pi,\pi)$ is the identity on $S^{2}\times S^{2}$, so the action is not effective. By taking half of the sum and difference, we see that $(X_{1},Y_{1})=\frac{1}{2}(\ell_{12}+\ell_{34},\ell_{12}-\ell_{34})$ is faithful with $2\pi$ period giving us a toric system on $S^{2}\times S^{2}$. The image of the momentum map of $(X_{1},Y_{1})$ is the unit square - the standard Delzant polytope for $S^{2}\times S^{2}$.

Note that the torus action of $(\ell_{12},\ell_{34})$ is effective on $T^{*}S^{3}$ as $(t_{12},t_{34})=(\pi,\pi)$ gives $\bm{x}\mapsto-\bm{x}$ and $\bm{y}\mapsto-\bm{y}$. However, the points $(\bm{x},\bm{y})$ and $(-\bm{x},-\bm{y})$ are anti-podal points on the same great circle and thus become the same point on $S^{2}\times S^{2}$ after reduction. ∎

7 Conclusion

The main novelty in this paper is the construction of a natural family of integrable system on $S^{2}\times S^{2}$ in section 4, and the analysis of its Liouville foliation in section 6. It turns out that many properties of the reduced system are visible already in one way or another in the original Stäckel system on $T^{*}S^{3}$. However, it should be pointed out that the upstairs system does not even have a natural Liouville foliation because it is superintegrable, and hence dynamically does not possess invariant tori, but just periodic orbits. After reduction by the flow of the Hamiltonian, which after extracting the square root is a global $S^{1}$ action, an integrable system on $S^{2}\times S^{2}$ is obtained. The reduced system is Lie-Poisson with Lie-algebra $\mathfrak{so}(4)$.

Since the reduction is done by the flow of the Hamiltonian the reduced system does not have a Hamiltonian any more, it just has commuting integrals. The definition of Liouville integrable system does not require a Hamiltonian, and the foliation into tori as defined by the integrals is defined independently of a Hamiltonian. What is missing is the possibility to define the Hamiltonian vector field which induces a flow on these tori. But this is not necessary in order to study the equivalence of Liouville foliations of integrable systems.

At first it may be surprising that in the Liouville-Arnold theorem the existence of action-angle variables near a regular torus does not need a Hamiltonian either. In fact, the action-angle variables are such that all of the integrals can be expressed as functions of the action variables. Moreover, considering diffeomorphism of the integrals changes the integrals, but does not change the action variables. We saw this explicitly for a restricted class of transformations of the integrals in our case. Since after reduction by the Hamiltonian there is no distinguished function any more, the focus is fully on the action variables. For the foliation it makes sense to consider leaf-preserving homeomorphisms or diffeomorphisms, but from the point of view of the action variables the natural class is symplectomorphisms. Since the reduced symplectic manifold is compact the image of the momentum map is compact as well, and we have shown that the image of the action map (appropriately modified so that it is continous!) is a right triangle. This triangle is rigid, which means that it is the same for the whole family. What does change are the position and organisation of action values in the triangle that correspond to critical values of the momentum map. These play the role of the height invariant, and in fact for the prolate system which is semi-toric these turn into the height invariant.

The Liouville-Arnold theorem holds near regular tori, and can be extended to open subsets of phase space bounded by separatrices. Only in rare cases are there no separatrices, essentially this means that the system is toric. But most integrable systems do have singular fibres that are not just tori, and the classification of integrable system needs to take these into account. It is crucial to note that the actions of the action-angle variables can in general not be extended globally in phase space. If this is possible we call them global $S^{1}$ actions. Instead of $S^{1}$ action we may also speak of a global $SO(2)$ symmetry. A slightly less optimal situation occurs for a global $SO(3)$ symmetry, which leads to almost global $S^{1}$ actions, as described for the Lamé system, with an almost global $S^{1}$ action and a spherical type singularity. Examples of global $S^{1}$ actions do appear in our family through degenerations, and when they do appear they unfold the action map into the polygon invariant in the semi-toric case (prolate system) and into the Delzant polygon in the toric case (cylindrical system).

Thus, for our family we have some analogues of important symplectic invariants, namely a convex polygon and generalisations of the height invariant. Certainly the semi-global symplectic invariants would need to be added, and at least in principle this is understood for the hyperbolic-hyperbolic point in the ellipsoidal family [14], and generalisations to elliptic-hyperbolic points, degenerate points and the rank 1 hyperbolic lines would need to be worked out. The interesting question is what kind of global invariants (like the twisting index invariant for semi-toric systems) would need to be added to the list so that it becomes the complete list of global symplectic invariants.

Appendix A Appendix

A.1 Identification of $\widetilde{Gr}(2,4)$ with $S^{2}\times S^{2}$

There is a natural identification of $\widetilde{Gr}(2,4)$ with $S^{2}\times S^{2}$ via the Plücker embedding

[TABLE]

If $(\bm{v}_{1},\bm{v}_{2},\bm{v}_{3},\bm{v}_{4})$ is an ordered-orthonormal basis of $\mathbb{R}^{4}$ then

[TABLE]

is a basis of $\Lambda^{2}(\mathbb{R}^{4})$ and a $\bm{L}\in\Lambda^{2}(\mathbb{R}^{4})$ is of the form

[TABLE]

The image of $\widetilde{Gr}(2,4)$ under $i$ is totally decomposable, i.e. any $\bm{L}\in\widetilde{Gr}(2,4)$ can be expressed as $L=\bm{x}\wedge\bm{y}$ where $\bm{x},\bm{y}\in\mathbb{R}^{4}$. Thus, $\bm{L}\in\text{im}(i)$ if and only if $\bm{L}\wedge\bm{L}=0$. This yields the Plücker relation

[TABLE]

This relation defines the image of $\widetilde{Gr}(2,4)$ under $i$. On $\Lambda^{2}(\mathbb{R}^{4})$ the Hodge star operator gives a decomposition

[TABLE]

where $\Lambda_{\pm}^{2}(\mathbb{R}^{4})$ are the $\pm 1$ eigenspaces. For convenience we let $\bm{V}_{ij}\coloneqq\bm{v}_{i}\wedge\bm{v}_{j}$. Then the effect of $\star$ on the basis vectors (56) is as follows

[TABLE]

This gives the following bases of $\Lambda_{+}^{2}(\mathbb{R}^{4})$ and $\Lambda_{-}^{2}(\mathbb{R}^{4})$:

[TABLE]

The Hodge star operator then induces a decomposition

[TABLE]

where $S_{+}^{2}\subseteq\Lambda_{+}^{2},S_{-}^{2}\subseteq\Lambda_{-}^{2}$. Note that

[TABLE]

are are both unit vectors in $\mathbb{R}^{3}$ with the bases (58) of $\Lambda^{2}_{\pm}$. Thus, a plane spanned by $(\bm{x},\bm{y})$ is identified with two unit vectors $(\bm{X},\bm{Y})$ in $\mathbb{R}^{3},$ i.e $S^{2}\times S^{2}$. Geometrically $S_{+}^{2}$ and $S_{-}^{2}$ represent the left and right isoclinic rotations in $\mathbb{R}^{4}$. Note that $\exp(\bm{x}\wedge\bm{y})$ is a rotation in the plane spanned by $\bm{x}$ and $\bm{y}$.

For a given point $(\bm{X},\bm{Y})\in S^{2}\times S^{2}$, we have $T^{-1}(\bm{X},\bm{Y})=(\ell_{12},\ell_{13},\ell_{14},\ell_{23},\ell_{24},\ell_{34})$ defines an element $\bm{L}\in\Lambda^{2}(\mathbb{R}^{4})$ where the map $T$ is defined in (20). For a given $\bm{L}$ we can find the corresponding circle on $S^{3}$ by considering the linear map

[TABLE]

Since $\bm{L}$ is decomposable with $\bm{L}=\bm{x}\wedge\bm{y}$ then $\ker(M_{\bm{L}})=\text{span}(\bm{x},\bm{y})$. Let $(b_{1},b_{2},b_{3},b_{4})$ be a basis of $\mathbb{R}^{4}$ and $(b_{123},b_{124},b_{134},b_{234})$ be bases of $\Lambda^{3}(\mathbb{R}^{4})$. Then we can represent $M_{\bm{L}}$ as

[TABLE]

If $\bm{L}$ is in the image of the Plücker embedding then $\text{rank}(M_{\bm{L}})$ is exactly $2$ as all $16$ of the $3\times 3$ minors vanishes under the Plücker relation. Thus, finding the nullspace of $M_{\bm{L}}$ gives us an unoriented plane in $\mathbb{R}^{4}$ whose intersection with $S^{3}$ is the fibre over $S^{2}\times S^{2}$.

A.2 Geodesic Flow on $S^{2}$

Following a similar construction to Section 3, we can define the geodesic flow on $S^{2}$ as a constrained system on $T^{*}\mathbb{R}^{3}$ with Cartesian coordinates $(\bm{x},\bm{y})$ where $\bm{x}\cdot\bm{x}=1$ and $\bm{x}\cdot\bm{y}=0$. The Dirac bracket yields a Poisson structure that can still be accurately represented by the matrix $B_{(\bm{x},\bm{y})}$ in (7), where here $\bm{x}$ and $\bm{y}$ are vectors in $\mathbb{R}^{3}$. The invariants of the Hamiltonian $H=\frac{1}{2}\bm{y}\cdot\bm{y}$ are the three angular momenta $\bm{L}=(\ell_{12},\ell_{13},\ell_{23})$. Symplectic reduction by the $S^{1}$ action of $\sqrt{2H}$ gives a reduced space that is diffeomorphic to the sphere $S^{2}$ given by the Casimir $\ell_{12}^{2}+\ell_{13}^{2}+\ell_{23}^{2}=2h=1$. The Poisson algebra of the invariants $\ell_{ij}$ is isomorphic to $\mathfrak{so}(3)$.

There are 2 non-equivalent orthogonal separable coordinates on the sphere $S^{2}$ leading to 2 distinct Stäckel systems. They are the spherical coordinates and the elliptic coordinates with semi-axes $(e_{1},e_{2},e_{3})$.

A.2.1 Elliptic Coordinates on $S^{2}$

The elliptic coordinates with semi-axes $(e_{1},e_{2},e_{3})$ are defined by

[TABLE]

Performing Stäckel separation or analysis using compatible Poisson structures both gives the separation constants $1=\ell_{12}^{2}+\ell_{13}^{2}+\ell_{23}^{2}$ and $\eta_{1}=e_{3}\ell_{12}^{2}+e_{2}\ell_{13}^{2}+e_{1}\ell_{23}^{2}$. Using the matrix $C=\text{diag}(\lambda-c_{1},\lambda-c_{2},\lambda-c_{3})$, the Poisson matrix for $(so_{3}^{*},\{\cdot,\cdot\}_{C})$ is given by

[TABLE]

The bracket $\{\cdot,\cdot\}_{C}$ drops rank only when exactly 2 of the $\ell_{ij}^{\prime}s$ vanishes, giving the 6 poles $\pm(1,0,0),\,\pm(0,1,0)$ and $\pm(0,0,1)$ as critical points on the sphere $S^{2}$ defined by fixing the Casimir. The image of the momentum map is the line segment $[e_{1},e_{3}]$ with 3 critical values at $e_{1},\,e_{2}$ and $e_{3}$. The critical points at $e_{1}$ and $e_{3}$ are elliptic and ones at $e_{2}$ are hyperbolic. The fibres of $e_{1}$ and $e_{3}$ are the poles $\pm(0,0,1)$ and $\pm(1,0,0)$ respectively. The preimage of $e_{2}$ is the intersection of the sphere $1=\ell_{12}^{2}+\ell_{13}^{2}+\ell_{23}^{2}$ with the ellipsoid $e_{2}=e_{3}\ell_{12}^{2}+e_{2}\ell_{13}^{2}+e_{1}\ell_{23}^{2}$. The sphere and this ellipsoid intersect tangentially at the poles $\pm(0,1,0)$.

The reduced system is of course the Euler top with phase space $S^{2}$ and Hamiltonian $\eta_{1}$, where $(e_{1},e_{2},e_{3})$ are the inverse moments of inertia of the top.

A.2.2 Spherical Coordinates on $S^{2}$

We define the spherical coordinates on $S^{2}$ with

[TABLE]

This system easily separates with separation constants $1=\ell_{12}^{2}+\ell_{13}^{2}+\ell_{23}^{2}$ and $\eta_{1}=\ell_{23}^{2}$. The image of the momentum map is the line segment $[0,1]$ with 2 critical values at [math] and $1$. The critical points at $1$ are the poles $\pm(0,0,1)$ and are elliptic. The point $\ell_{23}^{2}=0$ is degenerate and it’s fibre is the equator of the sphere.

The reduced system is the symmetric Euler top with phase space $S^{2}$ and two equal moments of inertia.