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In the first part of this paper, we give a global description of simplyconnected maximal Lorentzian surfaces whose group of isometries is of dimension1 (i.e. with a complete Killing field), in terms of a 1-dimensional generallynon-Hausdorff manifold (the space of Killing orbits) and a smooth functiondefined there. In the second part, we study the completeness of such surfacesand prove in particular that under the hypothesis of bounded curvature,completeness is equivalent to null completeness. We also give completenesscriterions involving the topological structure of the space of Killing orbits,or both the topology of this space and the geometry of the surface.

Maximal simply connected Lorentzian surfaces with a Killing field and their completeness