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The motion of charged particles in spacetimes containing a submanifold ofconstant positive or negative curvature is considered, with the electromagnetictensor proportional to the volume two-form form of the submanifold. In thepositive curvature case, this describes spherically symmetric spacetimes with amagnetic monopole, while in the negative curvature case, it is a hyperbolicspacetime with magnetic field uniform along hyperbolic surfaces. Constants ofmotion are found by considering Poisson brackets defined on a phase space withgauge-covariant momenta. In the spherically-symmetric case, we find acorrespondence between the trajectories on the Poincar\'{e} cone withequatorial geodesics in a conical defect spacetime. In the hyperbolic case, theanalogue of the Poincar\'{e} cone is defined as a surface in an auxiliaryMinkowski spacetime. Explicit examples are solved for the Minkowski,$\mathrm{AdS}_4\times S^2$, and the hyperbolic AdS-Reissner--Nordstr\"{o}mspacetimes.

Motion of charged particles in spacetimes with magnetic fields of spherical and hyperbolic symmetry