  In this work we find all helicoidal surfaces in Minkowski space with constant
mean curvature whose generating curve is a the 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 that the surface is ruled. If the generating curve
is a Lorentzian circle, we show that the only possibility is that the axis is
spacelike and the center of the circle lies in the axis.
