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

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