  We study the conformal classes of 2-dimensional Lorentzian tori with (non
zero) Killing fields. We define a map that associate to such a class a vector
field on the circle (up to a scalar factor). This map is not injective but has
finite dimensional fiber. It allows us to characterize the conformal classes of
tori with Killing field satisfying a condition related to the existence of
conjugate points given by L. Mehidi.
