  By using, the Vlasov-Poisson equation defined in either a Riemannian or a
semi-Riemannian space $\mathbb{R}^k_g$, and a Dirac distribution function, we
re-obtain the well known and classical equations of motion of a mechanical
system with a pairwise acting potential function. We apply this result to the
study of an $n$--body problem in a two dimensional negative space form with the
hyperbolic cotangent potential. Following the Klein's geometric Erlangen
program, with methods of M\"{o}bius geometry and using the Iwasawa
decomposition of the M\"{o}bius isometric group $SL(2,\mathbb{R})$ via its
representation in one Clifford Algebra, we complete the study of the whole set
of M\"{o}bius solutions (relative equilibria) of the problem begun by Diacu
{\it et al.} in \cite{Diacu8}.
