Superintegrable Bertrand Natural Mechanical Systems

The problem of finding superintegrable systems (i.e., the systems with closed trajectories in a certain domain) in the class of natural mechanical systems invariant under rotations goes back to the works of Bertrand and Darboux. The systems of Bertrand type under various restrictions were described by J. Bertrand, G. Darboux, V. Perlik, A. Besse, O. A. Zagryadsky, E. A. Kudryavtseva, and D. A. Fedoseev. However, in full generality, this issue remained open because of the so-called equator problem. In the remaining difficult case with equators, we describe all Bertrand's natural mechanical systems and also solve the problem on the connection between various classes of systems of Bertrand type (the widest class of locally Bertrand systems, the class of Bertrand systems, the narrow class of strongly Bertrand systems, and so on), which coincide in the previously studied case of configuration manifolds without equators. In particular, we show that strongly Bertrand systems form a meager subset in the set of Bertrand systems and Bertrand systems form a meager subset in the set of locally Bertrand systems.