Prescribed energy periodic solutions of Kepler problems with relativistic corrections

We consider two different relativistic versions of the Kepler problem in the plane: The first one involves the relativistic differential operator, and the second one involves a correction for the usual gravitational potential due to Levi-Civita. When a small external perturbation is added into such equations, we investigate the existence of periodic solutions with prescribed energy bifurcating from periodic invariant tori of the unperturbed problems. Our main tool is an abstract bifurcation theory from periodic manifolds developed by Weinstein, which is applied in the case of nearly integrable Hamiltonian systems satisfying the usual KAM isoenergetic nondegeneracy condition.