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Using twistor methods, we explicitly construct all local forms offour--dimensional real analytic neutral signature anti--self--dual conformalstructures $(M,[g])$ with a null conformal Killing vector. We show that $M$ isfoliated by anti-self-dual null surfaces, and the two-dimensional leaf spaceinherits a natural projective structure. The twistor space of this projectivestructure is the quotient of the twistor space of $(M,[g])$ by the group actioninduced by the conformal Killing vector.  We obtain a local classification which branches according to whether or notthe conformal Killing vector is hyper-surface orthogonal in $(M, [g])$. We giveexamples of conformal classes which contain Ricci--flat metrics on compactcomplex surfaces and discuss other conformal classes with no Ricci--flatmetrics.Comment: 43 pages, 4 figures. Theorem 2 has been improved: ASD metrics are  given in terms of general projective structures without needing to choose  special representatives of the projective connection. More examples (primary  Kodaira surface, neutral Fefferman structure) have been included. Algebraic  type of the Weyl tensor has been clarified. Final version, to appear in  Commun Math Phy

Anti-self-dual Conformal Structures with Null Killing Vectors from Projective Structures