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In this paper we study hypersurfaces $M_s^n$
{\it in} $\Bbb{R}_1^{n+1}$ ({\it or in} $\Bbb{R}^{n+1}$) verifying the equation $\Delta x= Ax+B$ and the condition that the principal
curvatures of the surface is not $(-n\in )$ times the mean curvature at the points where the mean curvature is nonzero.\\ \ \ \ \ We prove that the mean curvature of the surface is constant and as a result, either $M_s^n$ has zero mean curvature or when $M_s^n$ has at most two different principal curvatures, it is isoparametric. Here $M_s^n$ is a (pseudo) Riemannian manifold with metric of signature $s,s=0,1,\Bbb{R}_1^{n+1}$ is the $(n+1)$-dimensional flat Lorentzian space, $A$ is an endomorphism of $\Bbb{R}_1^{n+1}$ and $B\in \Bbb{R}_1^{n+1}$, $\Delta$ is the Laplacian operator on $M_s^n, x: M_s^n \longrightarrow \Bbb{R}_1^{n+1}$ is an isometric immersion.
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