From oscillator(s) and Kepler(s) potentials to general superintegrable systems in spaces of constant curvature

This paper presents a new viewpoint on the classification of quadratically superintegrable potentials in a space S2[k1]k2 of constant curvature k1 and metric signature (+, k2). Taking k1, k2 as parameters we deal at once with the spaces S2, H2, AdS1+1, dS1+1. All superintegrable potentials in any such space turn out to be related to four basic ones—oscillator, Kepler, half-oscillator and half-Kepler—by means of either a linear combination, a limiting procedure or a T-symmetry transformation. As compared with other ways to approach classification of superintegrable systems, we avoid direct computations as much as possible, yet we state complete results. Emphasis is on the interpretation and the mutual relationships between these systems.