M\"obius solutions of the curved $n$--body problem for positive curvature

We denote by $\mathbb{M}^2_R$ a two dimensional space of constant positive Gaussian curvature. With methods of M\"obius geometry and using the classification of the M\"obius group of automorphisms ${\rm \bf Mob}_2 \, (\hat{\mathbb{C}})$ of the Riemman sphere $\hat{\mathbb{C}}=\mathbb{M}_R^2 \cup \{\infty\}$, we give algebraic conditions for the existence of M\"obius solutions on $\hat{\mathbb{C}}$, getting a complete classification of them. We show several families of this kind of solutions.