  We are studying the harmonic and twistor equation on Lorentzian surfaces,
that is a two dimensional orientable manifold with a metric of signature
$(1,1)$. We will investigate the properties of the solutions of these equations
and try to relate the conformal invariant dimension of the space of harmonic
and twistor spinors to the natural conformal invariants given by the Lorentzian
metric. We will introduce the notion of semi-conformally flat surfaces and
establish a complete classification of the possible dimensions for this family.
