  Given a smooth function $U(t,x)$, $T$-periodic in the first variable and
satisfying $U(t,x) = \mathcal{O}(\vert x \vert^{\alpha})$ for some $\alpha \in
(0,2)$ as $\vert x \vert \to \infty$, we prove that the forced Kepler problem
$$ \ddot x = - \dfrac{x}{|x|^3} + \nabla_x U(t,x),\qquad x\in {\mathbb{R}}^2,
$$ has a generalized $T$-periodic solution, according to the definition given
in the paper [Boscaggin, Ortega, Zhao, \emph{Periodic solutions and
regularization of a Kepler problem with time-dependent perturbation}, Trans.
Amer. Math. Soc, 2018]. The proof relies on variational arguments.
