$\ast$-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.