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Conditions for the existence of periodic solutions to integrable two-dimensional Hamiltonian systems.

An apparent connection between integrability and the existence of a periodic solution of the variational equations around the straight-line solutions of planar Hamiltonian systems has been suggested [F. T. Hioe, Phys. Rev. A 39, 2628 (1989)]. Such solutions always exist if the gradient of the second integral and the Hamiltonian are independent on the straight lines. In any other case, the existence of such periodic solutions can be shown only if the second integral satisfies certain conditions.