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The solvability of the following system of difference equations $$z_{n+1}=alpha z_{n}^{a}w_{n}^{b},qquad w_{n+1}=beta w_{n-2}^{c}z_{n-2}^{d},quad nin {mathbb {N}}_{0}, $$where (a,b,c,din {mathbb {Z}}), (alpha ,beta , w_{-2}, w_{-1}, w_{0}, z_{-2}, z_{-1}, z_{0}in {mathbb {C}}setminus{0}), is studied in detail by using several methods. The system has the most complex structure of solutions of all the related systems studied so far, and some of the forms of solutions appear for the first time.

Product-type system of difference equations with a complex structure of solutions