On Lax representations under the gauge equivalence relation and Miura-type transformations for lattice equations

We study matrix Lax representations (MLRs) for differential-difference (lattice) equations. For a given equation, two MLRs are said to be gauge equivalent if one of them can be obtained from the other by means of a matrix gauge transformation. We present results on the following questions: 1. When is a given MLR gauge equivalent to an MLR suitable for constructing differential-difference Miura-type transformations by the method of [G. Berkeley, S. Igonin, J. Phys. A (2016), arXiv:1512.09123]? 2. When is a given MLR gauge equivalent to a trivial MLR? Furthermore, we present new examples of integrable differential-difference equations with Miura-type transformations.