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

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