The motion of charged particles in spacetimes containing a submanifold of constant positive or negative curvature are considered, with the electromagnetic tensor proportional to the volume two-form form of the submanifold. In the positive curvature case, this describes spherically symmetric spacetimes with a magnetic monopole, while in the negative curvature case, it is a hyperbolic spacetime with magnetic field uniform along hyperbolic surfaces.Constants of motion are found by considering Poisson brackets defined on a phase space with gauge-covariant momenta. In the spherically-symmetric case, we find a correspondence between the trajectories on the Poincaré cone with equatorial geodesics in a conical defect spacetime. In the hyperbolic case, the analogue of the Poincaré cone is defined as a surface in an auxiliary Minkowski spacetime. Explicit examples are solved for the Minkowski, AdS 4 × S 2 , and the hyperbolic AdS-Reissner-Nordström spacetimes.