  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.
