  It was demonstrated recently [Dunajski, Ferapontov and Kruglikov (2014)] that
the Manakov-Santini system describes a local form of general Lorentzian
Einstein-Weyl geometry. We introduce integrable matrix extension of the
Manakov-Santini system and show that it describes (2+1)-dimensional integrable
chiral model in Einstein-Weyl space. We develop a dressing scheme for the
extended MS system and define an extended hierarchy. Matrix extension of Toda
type system connected with another local form of Einstein-Weyl geometry is also
considered.
