$N(\kappa )$-paracontact metric manifolds admitting the Fischer-Marsden conjecture

We characterize $N(\kappa )$-paracontact metric manifolds (NKPMM) $M^{2n+1}$ satisfying the Fischer-Marsden conjecture. We demostrate that, if an $M^{2n+1}$ satisfies the Fischer-Marsden equation, then either $M^{2n+1}$ with $\kappa >-1$ is a non-Einstein manifold or $M^{2n+1}$ is locally isometric to ${\mathbb{E}}^{n+1}\times {\mathbb{H}}^n(-4)$ for $n>1$. For the $3$-dimensional case, we show that $M^3$ is an Einstein manifold.