  We study Lorentz hypersurfaces $M_{1}^{n}$ in $E_{1}^{n+1}$ satisfying
$\triangle \vec {H}= \alpha \vec {H}$ with non diagonal shape operator, having
complex eigenvalues. We prove that every such Lorentz hypersurface in
$E_{1}^{n+1}$ having at most five distinct principal curvatures has constant
mean curvature.
