Maximal surfaces in Lorentzian Heisenberg space

We build the twin correspondence between surfaces of constant mean curvature in R3 and maximal surfaces in the Lorentzian Heisenberg space Nil13(τ). We prove that Gauss maps of the maximal surfaces in Nil13(τ) are harmonic maps into S2 and that some perturbed Hopf differentials on the maximal surfaces in Nil13(τ) are holomorphic. We solve the Calabi-Bernstein problem for maximal surfaces in Nil13(τ).