Periodic solutions to a perturbed relativistic Kepler problem

We consider the perturbed relativistic Kepler problem d/dt ( m x' / sqrt{1-|x'|^2/c^2} ) = -α x / |x|^3 + ε ∇_x U(t,x), x ∈ R^2 {0}, where m, α > 0, where c is the speed of light, and U(t,x) is a function T-periodic in the first variable. For ε > 0 sufficiently small, we prove the existence of T-periodic solutions with prescribed winding number, bifurcating from invariant tori of the unperturbed problem