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

We denote by ${\mathbb{M}}_R^2$ 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}$})$oftheRiemmansphere$\hat{$\mathbb{C}$}=$\mathbb{M}$_R^2 \cup {\infty}$, we give algebraic conditions for the existence of M"obius solutions on $\widehat{\mathbb{C}}$, getting a complete classification of them. We show several families of this kind of solutions.