Compact Einstein-Weyl four-dimensional manifolds

We look for four-dimensional Einstein-Weyl spaces equipped with a regular Bianchi metric. Using the explicit four-parameter expression of the distance obtained in a previous work for non-conformally Einstein Einstein-Weyl structures, we show that only four one-parameter families of regular metrics exist on orientable manifolds: they are all of Bianchi type IX and locally conformally Kahler; moreover, in agreement with general results, they have a positive-definite conformal scalar curvature. In a Gauduchon gauge, they are compact and we obtain their topological invariants. Finally, we compare our results to the general analyses of Madsen, Pedersen, Poon and Swann: our simpler parametrization allows us to correct some of their assertions.