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