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Superintegrable systems on sphere

A. V. Borisov·I. S. Mamaev
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Abstract

We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, 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 $L^3$.

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A. V. Borisov, I. S. Mamaev. 2005-04-07. Superintegrable systems on sphere. https://arxiv.org/abs/nlin/0504018

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