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Solution to the solvability problem for a class of product-type systems of difference equations is given by presenting explicit formulas for its solutions. In the main/most complicated cases the problem is solved by using two different methods. This is the last system, out of the three non-equivalent ones, for which an associated polynomial, which essentially determines the structure of the solutions, is of the third order, so that the paper finishes the study of this kind of two-dimensional systems.