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Partially superintegrable (indeed isochronous) systems are not rare. (English) Zbl 1069.70009
Shabat, A.B.(ed.) et al., New trends in integrability and partial solvability. Proceedings of the NATO Advanced Research Workshop, Cadiz, Spain, June 12–16, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1835-5/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 132, 49-77 (2004).
Summary: We call partially superintegrable (indeed isochronous) those dynamical systems all solutions of which are completely periodic with a fixed period (“isochronous”) in a part of their phase space, and we review a recently introduced trick that allows to manufacture many such systems. Several examples are discussed.
For the entire collection see [Zbl 1050.35003].

70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
70H12 Periodic and almost periodic solutions for problems in Hamiltonian and Lagrangian mechanics