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 $mathbb Ctimes mathbb R$. Further we give a different proof for the singular Bjorling 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.