Spinning test particles in the $\gamma$ space-time

We consider the motion of spinning particles in the field of a well known vacuum static axially-symmetric space-time, known as $\gamma$-metric, that can be interpreted as a generalization of the Schwarzschild manifold to include prolate or oblate deformations. We derive the equations of motion for spinning test particles by using the Mathisson-Papapetrou-Dixon (MPD) equations together with the Tulczyjew spin-supplementary condition (SSC), and restricting the motion to the equatorial plane. We determine the limit imposed by super-luminal velocity for the spin of the particle located at the innermost stable circular orbits (ISCO). We show that the particles on ISCO of the prolate $\gamma$ space-time are allowed to have nigher spin than the corresponding ones in the the oblate case. We determine the value of the ISCO radius depending on the signature of the spin-angular momentum, ${\rm s-L}$ relation, and show that the value of the ISCO with respect to the non spinning case is bigger for ${\rm sL} 0$. The results may be relevant for determining the properties of accretion disks and constraining the allowed values of quadrupole moments of astrophysical black hole candidates.