Matrix Extension of the Manakov-Santini System and an Integrable Chiral Model on an Einstein-Weyl Background

We introduce an integrable matrix extension of the Manakov-Santini system and show that it describes a (2+1) -dimensional integrable chiral model in the Einstein-Weyl space. We apply a dressing scheme for the extended Manakov-Santini system and define an extended hierarchy. We also consider a matrix extension of a Toda-type system associated with another local form of the Einstein-Weyl geometry.