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

## TL;DR

This paper investigates conjugated point solutions in the 2-body problem on a 2D sphere, introducing a modified potential that allows solutions through antipodal points as limits of relative equilibria, revealing new invariant solution behaviors.

## Contribution

It proposes a modified potential to analyze solutions passing through antipodal points, extending the understanding of equilibrium solutions on spherical surfaces.

## Key findings

- Solutions through antipodal points can be obtained as limits of relative equilibria.
- Modified potential avoids singularities at conjugated points.
- Solutions are invariant under Killing vector fields and are geodesic curves.

## Abstract

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.