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

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.