Binary Darboux transformations and $N$-wave systems in rings

The covariance theorems for elementary and binary Darboux transformations in rings are formulated and proved for generalized Zakharov–Shabat problems. The definition of the elementary Darboux transformation is extended to an arbitrary number of orthogonal idempotents. The binary transformation is defined as a sequence of elementary transformations for direct and conjugate problems. The heredity property for the reduction constraints is established for some $UV$ pairs in rings; hence, the transformation generates solutions and infinitesimal symmetries of the corresponding zero-curvature equations. The explicit expressions for the transformations, solitons, and infinitesimals are given in the general case and in physically significant cases of extended non-Abelian $N$-wave equations (with linear terms added).