  Matrix differential-difference Lax pairs play an essential role in the theory
of integrable nonlinear differential-difference equations. We present
sufficient conditions which allow one 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 of Miura type to
known equations with Lax pairs.
  Suppose that one has three (possibly multicomponent) equations $E$, $E_1$,
$E_2$, a (Miura-type) 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. We present
several new integrable equations connected by new discrete substitutions of
Miura type to known equations.
