Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

We classify all helicoidal non-degenerate surfaces in Minkowski space with constant mean curvature and constant Gauss curvature whose generating curve is a graph of a polynomial or a Lorentzian circle. In the first case we prove that the degree of the polynomial is 0 or 1 and the surface is ruled. If the generating curve is a Lorentzian circle, we prove that the only possibility is that the axis is spacelike and the center of the circle lies on the axis