Periodic orbits for an infinite family of classical superintegrable systems Fr´ ed´ erick Tremblay 1 , Centre de recherches math´ ematiques and D´ epartement de math´ ematiques et de statistique, Universit´ e de Montreal, C.P. 6128, succ. Centre-ville, Montr´ eal (QC) H3C 3J7, Canada, Alexander V. Turbiner 2 , Instituto de Ciencias Nucleares, UNAM, A.P. 70-543, 04510 M´ exico, and Pavel Winternitz 3 , Centre de recherches math´ ematiques and D´ epartement de math´ ematiques et de statistique, Universit´ e de Montreal, C.P. 6128, succ. Centre-ville, Montr´ eal (QC) H3C 3J7, Canada Abstract We show that all bounded trajectories in the two dimensional classical system with the potential V (r, ϕ)= ω 2 r 2 + αk 2 r 2 cos 2 kϕ + βk 2 r 2 sin 2 kϕ are closed for all integer and rational values of k. The period is T = π 2ω and does not depend on k. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable. 1 tremblaf@crm.umontreal.ca 2 turbiner@nucleares.unam.mx 3 wintern@crm.umontreal.ca arXiv:0910.0299v1 [math-ph] 2 Oct 2009