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 vanishing $1-$form $\tau$, satisfying $L_r x =U +b$ 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 linearized operator of the$(r+1)th-$mean curvature hypersurface for$r=0,...,n$. For $r=0$, $L_0=\Delta^\eta$ nothing but (pseudo-)Laplacian on $(M, g, N)$. We prove that lightcone $\Lambda_0^{n+1}$, cylinders $\Lambda_0^{m+1}\times\R^{n-m}$, $1\leq m\leq n-1$ $(r+1)-$maximal Monge are only UCC-normalized normalization $\tau$ above equation. case $U$ scalar $ \lambda I$, $\lambda\in\R$ hence whole $M$, show $\Delta^\eta =\lambda +b$, open pieces hyperplanes.