Summary: As we all know for a surface \(M\) and a point \(P\in M\) in Euclidean \(3\)-space, the Weingarten map of \(M\) on \(P\) is always diagonalizable. Also if \(H^2(P)- K(P) = 0\) then we say that \(P\) is an umbilical point of \(M\). Here \(H\) is the mean curvature and \(K\) is the Gauss curvature of the surface. If all \(P\in M\) is umbilical then \(M\) is an umbilical surface. In Lorentz \(3\)-space the situation is different. The equation of \(H^2(P)-K(P) = 0\) for a point \(P\in M\) doesn’t mean that \(P\) is an umbilical point and the Weingarten map of \(M\) on \(P\) can be diagonalizable. In this paper we find the surfaces with the equation \(H^2-K = 0\), whose generating curve is a graph of a polynomial under homothetic motion groups in Lorentz \(3\)-space. We show that, this equation is possible when the axis of the homothetic motion is timelike or spacelike and the degree of the polynomial is \(0\) or \(1\), so the surfaces with \(H^2-K = 0\) are ruled surfaces. Also, despite having the equation \(H^2(P)- K(P) = 0\) for some \(P\in M\), these points are not umbilical on \(M\).