In this paper, we study Lorentz hypersurface $M_{1}^{n}$ in $E_{1}^{n+1}$ satisfying $\triangle \vec {H}= \alpha \vec {H}$ with minimal polynomial $[(y-\lambda)^{2}+\mu^{2}](y-\lambda_{1})(y-\lambda_{n})$ having shape operator \eqref{2.11}. We prove that every such Lorentz hypersurface in $E_{1}^{n+1}$ having at most four distinct principal curvatures has a constant mean curvature.