On common solutions of Mathisson equations under different conditions

In the context of investigations of possible highly relativistic motions of a spinning particle in the gravitational field, which can be described by the Mathisson equations under different supplementary condition, we analyze the circular orbits in a Schwarzschild field. The very orbits most clearly demonstrate the effect of the gravitational spin-orbit interaction on the particle's motion. It is shown that the Mathisson equations under the Frenkel-Mathisson and Tulczyjew-Dixon conditions have the common solutions describing the highly relativistic circular orbits in the region $r=3M(1+\delta), |\delta|\ll 1$. These orbits essentially differ from the geodesic circular orbits in the same region, particularly by the value of the particle's energy on the corresponding orbits.