Simplifications of Lax pairs for differential-difference equations by gauge transformations and (doubly) modified integrable equations

Matrix differential-difference Lax pairs play an essential role in the theory of integrable nonlinear differential-difference equations. We present sufficient conditions for the possibility to simplify such a Lax pair by matrix gauge transformations. Furthermore, we describe a procedure for such a simplification and present applications of it to constructing new integrable equations connected by (non-invertible) discrete substitutions to known equations with Lax pairs. Suppose that one has three (possibly multicomponent) equations $E$, $E_1$, $E_2$, a discrete substitution from $E_1$ to $E$, and a discrete substitution from $E_2$ to $E_1$. Then $E_1$ and $E_2$ can be called a modified version of $E$ and a doubly modified version of $E$, respectively. We demonstrate how the above-mentioned procedure helps (in the considered examples) to construct modified and doubly modified versions of a given equation possessing a Lax pair satisfying certain conditions. The considered examples include scalar equations of Itoh-Narita-Bogoyavlensky type and $2$-component equations related to the Toda lattice. Several new integrable equations and discrete substitutions are presented.