  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.
