  We investigate the stability of test-particle equilibrium orbits in
axisymmetric, but otherwise arbitrary, gravitational and electromagnetic
fields. We extend previous studies of this problem to include a toroidal
magnetic field. We find that, even though the toroidal magnetic field does not
alter the location of the circular orbits, it enters the problem as a
gyroscopic force with the potential to provide gyroscopic stability. This is in
essence similar to the situation encountered in the reduced three-body problem
where rotation enables stability around the local maxima of the effective
potential. Nevertheless, we show that gyroscopic stabilization by a toroidal
magnetic field is impossible for axisymmetric force fields in source-free
regions because in this case the effective potential does not possess any local
maxima. As an example of an axisymmetric force field with sources, we consider
the classical problem of a rotating, aligned magnetosphere. By analyzing the
dynamics of halo and equatorial particle orbits we conclude that axisymmetric
toroidal fields that are antisymmetric about the equator are unable to provide
gyroscopic stabilization. On the other hand, a toroidal magnetic field that
does not vanish at the equator can provide gyroscopic stabilization for
positively charged particles in prograde equatorial orbits.
