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.
Categories: math.DS