Hyperbolic relative equilibria for the negative curved n-body problem

We consider the n-body problem defined on surfaces of constant negative curvature. For the five and seven body case with a symmetric configuration, we give conditions on initial positions for the existence of collinear hyperbolic relative equilibria, where here collinear means that the bodies are on the same geodesic. If they satisfy the above conditions, then we give explicitly the values of the masses in terms of the positions, such that they can lead to these type of relative equilibria. The set of parameters that lead to these type of solutions has positive Lebesgue measure. For the general case of n-equal masses, we prove the no-existence of hyperbolic relative equilibria with a regular polygonal shape. In particular the Lagrangian hyperbolic relative equilibria do not exist.