Isometric immersions of L2 into L4

We give a global Weierstrass representation for isometric immersions with flat normal bundle from domains of the Lorentz plane L2 into L4. This representation relies on the choice of two holomorphic data on a Riemann surface, and the integration of a hyperbolic linear differential system. As applications, we give classification results and construct complete examples in explicit coordinates by exact integration of the differential system for some particular choices of the holomorphic data.