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We show that the space of orthogonally separable coordinates on the sphere$S^3$ induces a natural family of integrable systems, which after symplecticreduction leads to a family of integrable systems on $S^2 \times S^2$. Thegeneric member of the family corresponds to ellipsoidal coordinates. We use thetheory of compatible Poisson structures to study the critical points andcritical values of the momentum map. Interesting structure arises because theellipsoidal coordinate system can degenerate in a variety of ways, and allpossible orthogonally separable coordinate systems on $S^3$ (includingdegenerations) have the topology of the Stasheff polytope $K^4$, which is apentagon. We describe how the generic integrable system degenerates, and howthe appearance of global $SO(2)$ and $SO(3)$ symmetries is the main featurethat organises the various degenerate systems. For the whole family we showthat there is an action map whose image is an equilateral triangle. When highersymmetry is present, this triangle ``unfolds'' into a semi-toric polygon (whenthere is one global $S^1$-action) or a Delzant polygon (when there are twoglobal $S^1$-actions). We believe that this family of integrable systems is anatural playground for theories of global symplectic classification ofintegrable systems.

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