LAUSR

This paper essentially presents the last and important steps in the study of (practical) solvability of two-dimensional product-type systems of difference equations of the following form z n = α z n - k a w n - l b , w n = β w n - m c z n - s d , n ∈ N 0 , where k , l , m , s ∈ N , a , b , c , d ∈ Z , and where α , β and the initial values are complex numbers. It is devoted to the most complex case which has not been considered so far (the case k = l = s = 1 and m = 3 ). Closed form formulas for solutions to the system are found in all possible cases. The structure of the solutions to the system is considered in detail. The following five cases: (1) b = 0 ; (2) c = 0 ; (3) d = 0 ; (4) a c ≠ 0 ; (5) a = 0 , b c d ≠ 0 , are considered separately. Some of the situations appear for the first time in the literature.