Three body relative equilibria on ${\mathbb{S}}^2$

We study relative equilibria ($RE$ in short) for three-body problem on ${\mathbb{S}}^2$, under the influence of a general potential which only depends on $\cos {\sigma }_{ij}$ where ${\sigma }_{ij}$ are the mutual angles among the masses. Explicit conditions for masses $m_k$ and $\cos {\sigma }_{ij}$ to form relative equilibrium are shown. Using the above conditions, we study the equal masses case under the cotangent potential. We show the existence of scalene and isosceles Euler $RE$, and isosceles and equilateral Lagrange $RE$.