$\eta$-Ricci solitons on para-Kenmotsu manifolds

In the context of paracontact geometry, $\eta$-Ricci solitons are considered on manifolds satisfying certain curvature conditions: $(\xi,\cdot)_{R}\cdot S=0$, $(\xi,\cdot)_{S}\cdot R=0$, $(\xi,\cdot)_{W_2}\cdot S=0$ and $(\xi,\cdot)_{S}\cdot W_2=0$. We prove that on a para-Kenmotsu manifold $(M,\varphi,\xi,\eta,g)$, the existence of an $\eta$-Ricci soliton implies that $(M,g)$ is quasi-Einstein and if the Ricci curvature satisfies $(\xi,\cdot)_{R}\cdot S=0$, then $(M,g)$ is Einstein. Conversely, we give a sufficient condition for the existence of an $\eta$-Ricci soliton on a para-Kenmotsu manifold.