Revisiting the two-body problem in Yukawa gravity and in a gravitational extension of the Buckingham potential

We revisit the Keplerian-like parametrization of the two-body problem in Yukawa gravity studied in the literature. Some inconsistencies, which spoil Bertrand's theorem, observed in the $\ensuremath{\eta}$ parametrization of the true anomaly $\ensuremath{\theta}$ and in the formulae for the pericenter's advance are resolved. Moreover, inspired in this kind of study, we couple the Buckingham potential, a variation of the Lennard-Jones intermolecular potential, with the gravitational Newtonian potential and find a Keplerian-like parametrization for the solution of the two-body problem in this sort of gravity. The outcomes for the advance of the pericenter in both types of gravity are corroborated by using the Landau and Lifshitz's method. We also tested the expressions thus obtained against Solar System and S2 star data. The result for both models is that while some deviation from general relativity (GR) is allowed, GR cannot be discarded by the current analysis.