  In this paper, we are concerned with the monotonic and symmetric properties
of convex solutions Monge-Amp\`ere systems for instance, considering
  \begin{equation*}
  \det(D^2u^i)=f^i(x,{\bf u},\nabla u^i), \ 1\leq i\leq m,
  \end{equation*} over unbounded domains of various cases, including the whole
spaces $\mathbb{R}^n$, the half spaces $\mathbb{R}^n_+$ and the unbounded tube
shape domains in $\mathbb{R}^n$. We obtain monotonic and symmetric properties
of the solutions to the problem with respect to the geometry of domains and the
monotonic and symmetric properties of right-hand side terms. The proof is based
on carefully using the moving plane method together with various maximum
principles and Hopf's lemmas.
