  We study the relation between the existence of null conformal Killing vector
fields and existence of compatible complex and para-hypercomplex structures on
a pseudo-Riemannian manifold with metric of signature (2,2). We establish first
the topological types of pseudo-Hermitian surfaces admitting a nowhere
vanishing null vector field. Then we show that a pair of orthogonal, pointwise
linearly independent, null, conformal Killing vector fields defines a
para-hyperhermitian structure and use this fact for a classification of the
smooth compact four-manifolds admitting such a pair of vector fields. We also
provide examples of neutral metrics with two orthogonal, pointwise linearly
independent, null Killing vector fields on most of these manifolds.
