(4,0) and (4,4) sigma models with a tri-holomorphic Killing vector

We analyze (4,0) supersymmetric σ-models on a four dimensional target space which possess one tri-holomorphic Killing vector which is also assumed to leave invariant the torsion. The problem is reduced to two stages: first finding “special” three dimensional Einstein-Weyl spaces, and second solving a monopole-like equation on the special Einstein-Weyl space. A new class of examples is constructed using as Einstein-Weyl geometry the Berger sphere which includes the round three sphere as a particular case. When the Einstein-Weyl geometry is taken to be the round three sphere we show that the corresponding (4,0) geometries can be lifted to (4,4) geometries with two sets of non-commuting hyper-complex structures.