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.