Normalized null hypersurfaces in the Lorentz-Minkowski space satisfying $L_r x =U x +b$

In the present paper, we classify all normalized null hypersurfaces $x: (M,g,N)\to\R^{n+2}_1$ endowed with UCC-normalization with vanishing $1-$form $\tau$, satisfying $L_r x =U x +b$ for some (field of) screen constant matrix $U\in \R^{(n+2)\times(n+2)}$ and vector$b\in\R^{n+2}_{1}$, where $L_r$ is the linearized operator of the$(r+1)th-$mean curvature of the normalized null hypersurface for$r=0,...,n$. For $r=0$, $L_0=\Delta^\eta$ is nothing but the (pseudo-)Laplacian operator on $(M, g, N)$. We prove that the lightcone $\Lambda_0^{n+1}$, lightcone cylinders $\Lambda_0^{m+1}\times\R^{n-m}$, $1\leq m\leq n-1$ and $(r+1)-$maximal Monge null hypersurfaces are the only UCC-normalized Monge null hypersurface with vanishing normalization $1-$form $\tau$ satisfying the above equation. In case $U$ is the (field of) scalar matrix $ \lambda I$, $\lambda\in\R$ and hence is constant on the whole $M$, we show that the only normalized Monge null hypersurfaces $x: (M,g,N)\to\R^{n+2}_1$ satisfying $\Delta^\eta x =\lambda x +b$, are open pieces of hyperplanes.