The solvability of the following system of difference equations zn+1=αznawnb,wn+1=βwn−2czn−2d,n∈N0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$z_{n+1}=\\alpha z_{n}^{a}w_{n}^{b},\\qquad w_{n+1}=\\beta w_{n-2}^{c}z_{n-2}^{d},\\quad n\\in {\\mathbb {N}}_{0}, $$\\end{document} 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.