Superintegrable Systems on Sphere

We consider various generalizations of the Kepler problem to three-dimensional sphere S, a compact space of constant curvature. These generalizations include, among other things, addition of a spherical analog of the magnetic monopole (the Poincaré–Appell system) and addition of a more complicated field, which itself is a generalization of the MICZ-system. The mentioned systems are integrable — in fact, superintegrable. The latter is due to the vector integral, which is analogous to the Laplace–Runge–Lenz vector. We offer a classification of the motions and consider a trajectory isomorphism between planar and spatial motions. The presented results can be easily extended to Lobachevsky space L. 1 The Kepler problem in R Consider the Kepler problem: a mass point (of unit mass, without losing in generality) moves in the Newtonian field of a fixed center; the intensity of the gravitational interaction γ is constant. In this problem, in addition to the integral of energy