Elementary, binary and Schlesinger transformations in differential ring geometry

Schlesinger transformations are considered as special cases of elementary Darboux transformations of an abstract Zakharov-Shabat operator analog and its conjugate in differential rings and modules. The respective x- and t-chains of the transformations for potentials are constructed. Transformations that are combinations of the elementary ones for the special choice of direct and conjugate problems (named as binary ones) are applied within some constraints setting (reductions) for solutions. The geometric, structures: Darboux surfaces, Bianchi-Lie formula for (nonabelian) rings are specified. The applications in spectral operator and soliton theories are outlined.