  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.
