  We look for four dimensional Einstein-Weyl spaces equipped with a regular
Bianchi metric. Using the explicit 4-parameters expression of the distance
obtained in a previous work for non-conformally-Einstein Einstein-Weyl
structures, we show that only four 1-parameter families of regular metrics
exist on orientable manifolds : they are all of Bianchi $IX$ type and
conformally K\"ahler ; moreover, in agreement with general results, they have a
positive definite conformal scalar curvature. In a Gauduchon's 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 parametrisation allows us to correct some of their assertions.
