Superintegrable systems on a sphere

We consider various generalizations of the Kepler problem to three-dimensional sphere S' (a compact space of constant curvature). In particular, these generalizations include addition of a spherical analogue of the magnetic monopole (the Poincare-Appell system) and addition of a more complicated field which is a generalization of the MICZ-system. The mentioned systems are integrable superintegrable, and there exists 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-3.