Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

We study the Jones and Tod correspondence between selfdual conformal 4-mani folds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shea r-free geodesic congruences. Such congruences exist in abundance and so pro vide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kahler metrics and hypercomplex struc tures with symmetry. We also show that in the presence of such a congruence , the Einstein-Weyl equation is equivalent to a pair of coupled monopole eq uations, and we solve these equations in a special case. The new Einstein-W eyl spaces, which we call Einstein-Weyl "with a geodesic symmetry", give ri se to hypercomplex structures with two commuting triholomorphic vector fiel ds.