$ \ast $-Ricci tensor on $ (\kappa, \mu) $-contact manifolds

We introduce the notion of semi-symmetric $ \ast $-Ricci tensor and illustrate that a non-Sasakian $ (\kappa, \mu) $-contact manifold is $ \ast $-Ricci semi-symmetric or has parallel $ \ast $-Ricci operator if and only if it is $ \ast $-Ricci flat. Then we find that among the non-Sasakian $ (\kappa, \mu) $-contact manifolds with the same Boeckx invariant $ I_M $, only one is $ \ast $-Ricci flat, so we can think of it as the representative of such class. We also give two methods to construct $ \ast $-Ricci flat $ (\kappa, \mu) $-contact manifolds.