Solvable product-type system of difference equations whose associated polynomial is of the fourth order

The solvability problem for the following system of difference equations Z(n+1) = alpha Z(n)(a)w(n)(b), w(n+1) = beta w(n-1)(c)z(n-2)(d), n is an element of N-0, where a, b, c, d is an element of Z, alpha, beta is an element of C \ {0}, z(-2), z(-1), z(0), w(-1), w(0) is an element of C \ {0}, is solved. In the main case when bd not equal 0, a polynomial of the fourth order is associated to the system, and its solutions are represented in terms of the parameters, through the roots of the polynomial in all possible cases (the roots are given in terms of parameters a, b, c, d). This is also the first paper which successfully deals with the associated polynomial (to a product-type system) of the fourth order in detail, which is the main achievement of the paper.