  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.
