The Drinfeld–Sokolov reduction for a Lax difference operator with the periodic boundary conditions in the case $gl\bigl(n,\mathbb C((\lambda^{-1}))\bigr)$

Matrix difference equations of a special kind are investigated by means of the dressing equations method. It is shown that these equations are gauge invariant, the corresponding flows being commutative. It is proved that the equation for the gauge equivalence class and the Lax equation are equivalent.