  We study the family of quantum integrable systems that arise from separating
the Schr\"odinger equation in all 6 separable orthogonal coordinates on the 3
sphere: ellipsoidal, prolate, oblate, Lam\'{e}, spherical and cylindrical. On
the one hand each separating coordinate system gives rise to a quantum
integrable system on S2 x S2, on the other hand it also leads to families of
harmonic polynomials in R4. We show that separation in ellipsoidal coordinates
yields a generalised Lam\'{e} equation - a Fuchsian ODE with 5 regular singular
points. We seek polynomial solutions so that the eigenfunctions are analytic at
all finite singularities. We classify eigenfunctions by their discrete symmetry
and compute the joint spectrum for each symmetry class. The latter 5 separable
coordinate systems are all degenerations of the ellipsoidal coordinates. We
perform similar analyses on these systems and show how the ODEs degenerate in a
fashion akin to their respective coordinates. For the prolate system we show
that there exists a defect in the joint spectrum which prohibits a global
assignment of quantum numbers: the system has quantum monodromy. This is a
companion paper to our previous work where the respective classical systems
were studied.
