Dispersionless integrable systems and the Bogomolny equations on an Einstein–Weyl geometry background

We obtain a dispersionless integrable system describing a local form of a general three-dimensional Einstein–Weyl geometry with a Euclidean (positive) signature, construct its matrix extension, and show that it leads to the Bogomolny equations for a non-Abelian monopole on an Einstein–Weyl background. We also consider the corresponding dispersionless integrable hierarchy, its matrix extension, and the dressing scheme.