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Nonminimal spin-gravity interaction through unit gravimagnetic moment leadsto modified Mathisson-Papapetrou-Tulczyjew-Dixon equations with improvedbehavior in the ultrarelativistic limit. We present exact Hamiltonian of theresulting theory and compute an effective $\frac{1}{c^2}$\,-Hamiltonian andleading post-Newtonian corrections to the trajectory and spin. Gravimagneticmoment causes the same precession of spin ${\bf S}$ as a fictitious rotation ofthe central body with angular momentum ${\bf J}=\frac{M}{m}{\bf S}$. So themodified equations imply a number of qualitatively new effects, that could beused to test experimentally, whether a rotating body in general relativity hasnull or unit gravimagnetic moment.

Frame-dragging effect in the field of non rotating body due to unit gravimagnetic moment