  A new representation, which does not contain the third-order derivatives of
the coordinates, of the exact Mathisson-Papapetrou-Dixon equations, describing
the motion of a spinning test particle, is obtained under the assumption of the
Mathisson-Pirani condition in a Kerr background. For this purpose the integrals
of energy and angular momentum of the spinning particle as well as a
differential relationship following from the Mathisson-Papapetrou-Dixon
equations are used. The form of these equations is adapted for their computer
integration with the aim to investigate the influence of the spin-curvature
interaction on the particle's behavior in the gravitational field without
restrictions on its velocity and spin orientation. Some numerical examples for
a Schwarzschild background are presented.
