  Using the Mathisson-Papapetrou-Dixon (MPD) equations, we investigate the
trajectories of a spinning particle starting near $r_{ph}^{(-)}$ in a Kerr
field and moving with the velocity close to the velocity of light
($r_{ph}^{(-)}$ is the Boyer-Lindquist radial coordinate of the
counter-rotation circular photon orbits). First, as a partial case of these
trajectories, we consider the equatorial circular orbit with $r=r_{ph}^{(-)}$.
This orbit is described by the solution that is common for the rigorous MPD
equations and their linear spin approximation. Then different cases of the
nonequatorial motions are computed and illustrated by the typical figures. All
these orbits exhibit the effects of the significant gravitational repulsion
that are caused by the spin-gravity interaction. Possible applications in
astrophysics are discussed.
