We examine the unboundedness, persistence, boundedness, uniqueness, and existence of non-negative equilibrium of an exponential symmetric difference equations system: ${\Omega }_{n+1}={\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)}$, ${\varpi }_{n+1}={\alpha }_2+{\beta }_2{\varpi }_n+{\gamma }_2{\varpi }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},$$n=0,1,\cdots$, whereby initial values ${\Omega }_{-1},{\varpi }_{-1},{\Omega }_0,{\varpi }_0$ and parameters ${\alpha }_1,{\alpha }_2$ are non-negative real numbers and ${\beta }_1,{\beta }_2\in (0,1)$ and ${\gamma }_1,{\gamma }_2\le 0$. We will discuss asymptotic global and local stability and the convergence rate of this system. Ultimately, to check our results, we set out some numerical explanations.