  Off-equatorial circular orbits with constant latitudes (halo orbits) of
electrically charged particles exist near compact objects. In the previous
paper, we discussed this kind of motion and demonstrated the existence of
minima of the two-dimensional effective potential which correspond to the
stable halo orbits. Here, we relax previous assumptions of the pseudo-Newtonian
approach for the gravitational field of the central body and study properties
of the halo orbits in detail. Within the general relativistic approach, we
carry out our calculations in two cases. Firstly, we examine the case of a
rotating magnetic compact star. Assuming that the magnetic field axis and the
rotation axis are aligned with each other, we study the orientation of motion
along the stable halo orbits. In the poloidal plane, we also discuss shapes of
the related effective potential halo lobes where the general off-equatorial
motion can be bound. Then we focus on the halo orbits near a Kerr black hole
immersed in an asymptotically uniform magnetic field of external origin. We
demonstrate that, in both the cases considered, the lobes exhibit two different
regimes, namely, one where completely disjoint lobes occur symmetrically above
and below the equatorial plane, and another where the lobes are joined across
the plane. A possible application of the model concerns the structure of
putative circumpulsar discs consisting of dust particles. We suggest that the
particles can acquire a small (but non-zero) net electric charge, and this
drives them to form the halo lobes.
