Reparametrizing the relativistic Kepler equation: a bridge to Levi-Civita-type models

We establish a link between different relativistic variants of the Kepler problem. In particular, we show that solutions of the special relativistic model with fixed energy can be reparameterized as solutions of a generalized Kepler equation with an additional $1\mathrm{/}{r}^2$ term in the gravitational potential. This yields a dynamics of the same type as the Levi-Civita correction, up to different coefficients.