  We study the motion of charged particles in a family of five-dimensional
solutions describing either a black hole or topological star with a fifth
compact dimension stabilised by a magnetic flux. The particle's trajectory is
shown to move along the surface of a Poincar\'{e} cone. The radial motion shows
a rich structure where the existence of various bound, plunging, or escaping
trajectories depend on theconstants of motion. Curves of energy and angular
momentum corresponding to spherical orbits show a swallow-tail structure highly
reminiscent to phase transitions of thermodynamics. When the momentum along the
compact direction is varied, the is a critical point beyond which the
swallow-tail kink disappears and becomes a smooth curve.
