Discrete Darboux system with self-consistent sources and its symmetric reduction

The discrete non-commutative Darboux system of equations with self-consistent sources is constructed, utilizing both the vectorial fundamental (binary Darboux) transformation and the method of additional independent variables. Then the symmetric reduction of discrete Darboux equations with sources is presented. In order to provide a simpler version of the resulting equations we introduce the $\tau/\sigma$ form of the (commutative) discrete Darboux system. Our equations give, in continuous limit, the version with self-consistent sources of the classical symmetric Darboux system.