  To model magnetic fields of compact objects we solve the Maxwell equations in
the background of the exterior static Schwarzschild and slowly rotating Kerr
space-times. We impose the boundary condition that the electromagnetic fields
are to vanish at infinity. A full set of solutions is obtained, describing
axially symmetric magnetic fields, supplemented by axial electric fields in the
case of non-vanishing rotation of the gravitational background. We study the
motion of charged test particles in these combined gravitational and
electromagnetic fields, in particular considering the conditions for circular
equatorial orbits. Such orbits always exist in odd-multipole magnetic fields,
and they can exist for particular radii in a combination of two or more
even-multipole magnetic fields. Combinations of several odd-multipole fields
can give rise to radial variation in the field orientation and the direction of
motion of charged particles. Deviations from circularity are described using a
perturbative approach. This also allows to study the stability of the parent
circular orbits.
