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

Matrix extension of the Manakov-Santini system and integrable chiral model on Einstein-Weyl background