  Near a black hole or an ultracompact star, 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 study a possibility that the interplay between
gravitational and electromagnetic action may allow for stable, energetically
bound off-equatorial motion of charged particles. This would represent
well-known generalized Stormer's 'halo' orbits, which have been discussed in
connection with the motion of dust grains in planetary magnetospheres. We
demonstrate 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. In
the case of Kerr-Newman solution, numerical analysis shows that the mutually
connected gravitational and electromagnetic 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.
