The problem of linear input-output (i/o) equivalence of meromorphic nonlinear control systems, described by implicit higher order difference equations, is studied. It is proved that any system is linearly i/o equivalent to a row-reduced form. The constructive algorithm is given for finding the required transformation. The latter amounts to 1) multiply the set of i/o equations phi = 0 from left by a unimodular matrix A(delta), whose entries are non-commutative polynomials in the forward-shift operator delta, and 2) define certain multiplicative subset of the difference ring of analytic functions which introduces some inequations that should be satisfied.