  We consider a finite-temperature Bose-Einstein condensate in a
quasi-two-dimensional trap containing a single precessing vortex. We find that
such a configuration arises naturally as an ergodic equilibrium of the
projected Gross-Pitaevskii equation, when constrained to a finite conserved
angular momentum. In an isotropic trapping potential the condensation of the
classical field into an off-axis vortex state breaks the rotational symmetry of
the system. We present a methodology to identify the condensate and the
Goldstone mode associated with the broken rotational symmetry in the
classical-field model. We also examine the variation in vortex trajectories and
thermodynamic parameters of the field as the energy of the microcanonical field
simulation is varied.
