  For the $n$-body problem in spaces of negative constant Gaussian curvature,
we prove for a class of negative hyperbolic rotopulsators that if that class
exists, the configurations of the point masses of these rotopulsators have to
be regular polygons if the rotopulsators are not relative equilibria.
Additionally, we prove that if the rotopulsators are relative equilibria, there
exists at most one such solution.
