Conjugated equilibrium solutions for the $2$--body problem in the two dimensional sphere $\mathbb{M}^2_R$ for equal masses

We study here the behaviour of solutions for conjugated (antipodal) points in the $2$-body problem on the two-dimensional sphere $\mathbb{M}^2_R$. We use a slight modification of the classical potential used commonly in \cite{Borisov}, \cite{Diacu} and \cite{Perez}, which avoids the conjugated (antipodal) points as singularities and permit us obtain solutions through these points, as limit of relative equilibria. Such limit solutions behave as relative equilibria because are invariant under Killing vector fields in the Lie Algebra ${\rm su} (2)$ and are geodesic curves.