  In this paper we investigate the three-dimensional (3D) motion of a test
particle in a stationary, axially symmetric spacetime around a central compact
object, under the influence of a radiation field. To this aim we extend the
two-dimensional (2D) version of the Poynting-Robertson effect in General
Relativity (GR) that was developed in previous studies. The radiation flux is
modeled by photons which travel along null geodesics in the 3D space of a Kerr
background and are purely radial with respect to the zero angular momentum
observer (ZAMO) frames. The 3D general relativistic equations of motion that we
derive are consistent with the classical (i.e. non-GR) description of the
Poynting-Robertson effect in 3D. The resulting dynamical system admits a
critical hypersurface, on which radiation force balances gravity. Selected test
particle orbits are calculated and displayed, and their properties described.
It is found that test particles approaching the critical hypersurface at a
finite latitude and with non-zero angular moment are subject to a latitudinal
drift and asymptotically reach a circular orbit on the equator of the critical
hypersurface, where they remain at rest with respect to the ZAMO. On the
contrary, test particles that have lost all their angular momentum by the time
they reach the critical hypersurface do not experience this latitudinal drift
and stay at rest with respects to the ZAMO at fixed non-zero latitude.
