The authors study pseudo-Riemannian submanifolds in \(R^{n+k}_ t\) with the condition \(\Delta x=Ax+B\) where \(A\) is an endomorphism of \(R^{n+k}_ t\) and \(B\) a constant vector in \(R^{n+k}_ t\). They give a characterization theorem when \(A\) is a self adjoint endomorphism of \(R^{n+k}_ t\) and a classification theorem for any endomorphism \(A\) in the case of hypersurfaces. In this case \(M^ n_ s\) satisfies \(\Delta x=Ax+B\) if and only if it is an open piece of a minimal hypersurface, a totally umbilical hypersurface or a pseudo-Riemannian product of a totally umbilical and a totally geodesic submanifold.