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We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension 3. We in particular obtain a representation theorem for surfaces in L (κ , τ) spaces. We then recover the Calabi correspondence between minimal surfaces in R 3 and maximal surfaces in R 1 3 , and obtain a new Lawson type correspondence between CMC surfaces in R 1 3 and in the 3-dimensional pseudo-hyperbolic space H 1 3. [ABSTRACT FROM AUTHOR]