  This work studies the symmetries, the associated momentum map, and relative
equilibria of a mechanical system consisting of a small axisymmetric magnetic
body-dipole in an also axisymmetric external magnetic field that additionally
exhibits a mirror symmetry; we call this system the ``orbitron". We study the
nonlinear stability of a branch of equatorial quasiorbital relative equilibria
using the energy-momentum method and we provide sufficient conditions for their
$\mathbb{T}^2$--stability that complete partial stability relations already
existing in the literature. These stability prescriptions are explicitly
written down in terms of the some of the field parameters, which can be used in
the design of stable solutions. We propose new linear methods to determine
instability regions in the context of relative equilibria that we use to
conclude the sharpness of some of the nonlinear stability conditions obtained.
