A class of integrable potentials

Abstract

A class of time independent two-dimensional integrable potentials, all possessing an invariant of the same general form, is constructed. One of these potentials is superintegrable, its invariants realize the symmetry algebra sO(3) for negative energies, e(2) for zero energy, and sO(2,1) for positive energies. A transformation of coupling constants reveals that in parabolic coordinates this potential is the harmonic oscillator acted on by constant forces. This and another potential in the class may be considered as successive extensions of the Kepler potential. The analytic properties of these integrable systems in the complex time plane are also discussed.