  We study the simply connected inextendable Lorentzian surfaces admitting a
Killing vector field. We construct a natural family of such surfaces, that we
call "universal extensions". They are characterized by a condition of symmetry,
the "reflexivity", and a by a rather weak completeness assumption, the absence
of "saddles at infinity". Considering these surfaces as model spaces, we study
their minimal quotients, divisible open sets and conjugate points. We show
uniformisation results (by an open subset of one of these universal extensions,
which is uniquely determined) in the following cases: compact surfaces and
analytical surfaces. It allows us to give a classification of Lorentzian tori
and Klein bottles with a Killing vector field.
