We construct explicitly, using the generalized Weierstrass representation, a complete embedded minimal surface M _k,θ invariant under a rotation of order k + 1 and a screw motion of angle 2θ about the same axis, where k > 0 is any integer and ois any angle with |θ| < π/(k + 1). The existence of such surfaceswas proved in [Callahan et al. 1990), but no practical procedure for constructing them was given there.We also show that the sameproblem for θ = ±π/(k+1) does not have a solution enjoying reflective symmetry; the question of the existence of a solution without such symmetry is left open.