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 $RE$.