  We demonstrate that the Schr\"odinger equation for two electrons on a ring,
which is the usual paradigm to model quantum rings, is solvable in closed form
for particular values of the radius. We show that both polynomial and
irrational solutions can be found for any value of the angular momentum and
that the singlet and triplet manifolds, which are degenerate, have distinct
geometric phases. We also study the nodal structure associated with these
two-electron states.

Strongly Correlated Electrons;Other Condensed Matter;Chemical Physics