Periodic solutions to a perturbed relativistic Kepler problem

We consider a perturbed relativistic Kepler problem \begin{equation*} \dfrac{$\mathrm{d}$}{$\mathrm{d}$t}\left(\dfrac{m\dot{x}}{\sqrt{1-|\dot{x}|^2/c^2}}\right)=-\alpha, \dfrac{x}{|x|^3}+\varepsilon , \nabla_x U(t,x), \qquad x \in $\mathbb{R}$^2 \setminus {0}, \end{equation*} where $m,\alpha >0$, $c$ is the speed of light and $U(t,x)$ is a function $T$-periodic in the first variable. For $\epsilon >0$ sufficiently small, we prove the existence of $T$-periodic solutions with prescribed winding number, bifurcating from invariant tori of the unperturbed problem.