  We study the motion of a spinning particle with gravimagnetic moment in
Schwarzschild-like spacetimes with a metric $ds^2=-f(r) dt^2 + f^{-1}(r) dr^2 +
r^2 d\Omega^2$, specifically we deal with Schwarzschild, Reissner-Nordstrom
black holes as well as Ayon-Beato-Garcia and Bardeen regular spacetimes. First,
we introduce the Hamiltonian system of equations which describes such kind of
particles. In the case of null gravimagnetic moment, the equations are
equivalent to the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations.
Working in the equatorial plane, using the constants of motion generated by the
symmetries of the considered spacetimes and the Spin Supplementary Conditions
(SSC), we change the problem of solving six differential equations for the
momenta and the non-vanishing spin-tensor components to solving six algebraic
equations. We show that the equation for the $P_0(r)$ component totally
decouples, $P_0(r)$ can be found by solving a 6th order polynomial. We analyze
the conditions for existence of solutions of this algebraic system for the
relevant cases of gravimagnetic moment equal to unit, which corresponds to a
gravimagnetic particle, and zero which corresponds to the MTPD system. A
numerical algorithm to generate solutions of the momenta $P_{\mu}$ is provided
and some solutions are generated.
