  We compute exact Hamiltonian (and corresponding Dirac brackets) for spinning
particle with gravimagnetic moment $\kappa$ in an arbitrary gravitational
background. $\kappa=0$ corresponds to the Mathisson-Papapetrou-Tulczyjew-Dixon
(MPTD) equations. $\kappa=1$ leads to modified MPTD equations with reasonable
behavior in the ultrarelativistic limit. So we study the modified equations in
the leading post-Newtonian approximation. Rotating body with unit gravimagnetic
moment has qualitatively different behavior as compared with MPTD body: A) If a
number of gyroscopes with various rotation axes are freely traveling together,
the angles between the axes change with time. B) For specific binary systems,
gravimagnetic moment gives a contribution to frame-dragging effect with the
magnitude, that turns out to be comparable with that of Schiff frame dragging.
