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We analyze one particle, two-center quantum problems which admit separationof variables in prolate spheroidal coordinates, a natural restriction satisfiedby the H$_2^+$ molecular ion. The symmetry operator is constructed explicitly.We give the details of the Hamiltonian reduction of the 3D system to a 2Dsystem with modified potential that is separable in elliptic coordinates. Thepotentials for which there is double-periodicity of the Schr\"odinger operatorin the space of prolate spheroidal coordinates, including one for the H$_2^+$molecular ion, are indicated. We study possible potentials that admitexact-solvability is as well as all models known to us with the(quasi)-exact-solvability property for the separation equations. We find deepconnections between second-order superintegrable and conformallysuperintegrable systems and these tractable problems. In particular we derive ageneral 4-parameter expression for a model potential that is always integrableand is conformally superintegrable for some parameter choices.

Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability