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In this paper, we construct and formulate the solutions and periodicity character of the following nonlinear rational systems of difference equations: $ \begin{equation} S_{n+1} = \dfrac{T_{n} S_{n-2}}{S_{n-2} + T_{n-1}},\quad T_{n+1} = \dfrac{S_{n} T_{n-2}}{\pm T_{n-2} \pm S_{n-1}},\quad \;\;\;\; n = 0,1,2,...., \quad\quad\quad(0.1)\end{equation} $ <p>where the initial conditions $ s_{-2}, s_{-1}, s_0, t_{-2}, t_{-1}, t_0 $ are positive real numbers. Moreover, some mathematical programs are used to support our theoretical results of each system in this paper.