Existence of maximal surface containing given curve and special singularity

We give a different formulation for describing maximal surfaces in Lorentz-Minkowski space, ${\mathbb{L}}^3$, using the identification of $\mathbb L^3$with$\mathbb C\times \mathbb R$. Further we give a different proof for the singular Bj"orling problem for the case of closed real analytic null curve. As an application, we show the existence of maximal surface which contains a given curve and has a special singularity.