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 $\varepsilon > 0$ sufficiently small, we prove the existence of $T$-periodic solutions with prescribed winding number, bifurcating from invariant tori of the unperturbed problem.