In this paper we deal with the uniqueness of the Helicoid and Enneper's surface as maximal immersions in the Lorentz-Minkowski space $\mathbb{R}^3_1$. In both cases the surface contains a proper lightlike arc where the immersion folds back, and so the image via the immersion is a (double) surface without selfintersections with lightlike boundary of mirror symmetry.