Summary: In this paper, we show that the system of difference equations \[ x_{n}=\frac{x_{n-2}y_{n-3}}{y_{n-1} (a_n+b_{n}x_{n-2}y_{n-3} )},\quad y_{n}=\frac{y_{n-2}x_{n-3}}{x_{n-1}(\alpha_n+\beta_{n}y_{n-2}x_{n-3} )},\quad n\in\mathbb{N}_0, \] where the sequences \(\forall n\in\mathbb{N}_0, (a_n),(b_n),(\alpha_n),(\beta_n)\) and the initial values \(x_{-j}, y_{-j}, j\in\{1,2,3\}\) are non-zero real numbers, can be solved in the closed form. For the case when all the sequences \((a_n),(b_n),(\alpha_n),(\beta_n)\) are constant we describe the asymptotic behavior and periodicity of solutions of above system is also investigated.