  We consider two different relativistic versions of the Kepler problem in the
plane: the first one involves the relativistic differential operator, 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 non-degeneracy condition.
