  We show that the exact wave function for two electrons, interacting through a
Coulomb potential but constrained to remain on the surface of a
$\mathcal{D}$-sphere ($\mathcal{D} \ge 1$), is a polynomial in the
interelectronic distance $u$ for a countably infinite set of values of the
radius $R$. A selection of these radii, and the associated energies, are
reported for ground and excited states on the singlet and triplet manifolds. We
conclude that the $\mathcal{D}=3$ model bears the greatest similarity to normal
physical systems.
