Periodic Solutions of Symmetric Perturbations of Gravitational Potentials

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It is shown that for certain symmetric perturbations of gravitational potentials in the space, which admit two first integrals of motion, a circular solution of the unperturbed system with inclination different from 0 and π gives rise to a periodic solution of the reduced dynamics which is defined in the quotient space of the action by the subgroup that fixes the symmetry axis. In the planar case, if we assume that the system admits a first integral of motion which is also symmetric with respect to the origin, then it is shown that each circular solution of the unperturbed problem gives rise to a periodic solution of the perturbed system.