Near black holes or neutron stars, motion of particles is governed by strong gravitational field. Electrically charged particles feel also electromagnetic force arising due to currents inside the star or plasma circling around. We present our study of a possibility that the interplay between gravitational and electromagnetic action may allow for stable, energetically bound off-equatorial motion of charged particles.
This represents well-known generalized St\"ormer's `halo orbits', which have been discussed in connection with the motion of dust grains in planetary magnetospheres. Investigating models of compact star and Schwarzschild (Kerr) black hole with current loop in the equatorial plane, we have demonstrated that such orbits exist and can be astrophysically relevant when a compact star or a black hole is endowed with a dipole-type magnetic field. However, in the case of Kerr-Newman spacetimes, numerical analysis shows that the mutually connected gravitational and electromagnetic (magnetic field of dipole-type) fields do not allow existence of stable halo orbits above the outer horizon of black holes. Such orbits are either hidden under the inner black-hole horizon, or they require the presence of a naked singularity.