  We investigate symmetry properties of positive solutions for fully nonlinear
uniformly elliptic systems, such as $$ F_i \,(x,Du_i,D^2u_i) +f_i \,(x,u_1,
\ldots , u_n,Du_i)=0, \;\; 1 \leq i \leq n, $$ in a bounded domain $\Omega$ in
$\mathbb{R}^N$ with Dirichlet boundary condition $u_1=\ldots,u_n=0$ on
$\partial\Omega$. Here, $f_i $'s are nonincreasing with the radius $r=|x|$, and
satisfy a cooperativity assumption. In addition, each $f_i $ is the sum of a
locally Lipschitz with a nondecreasing function in the variable $u_i$, and may
have superlinear gradient growth. We show that symmetry occurs for systems with
nondifferentiable $f_i$'s by developing a unified treatment of the classical
moving planes method in the spirit of Gidas-Ni-Nirenberg. We also present
different applications of our results, including uniqueness of positive
solutions for Lane-Emden systems in the subcritical case in a ball, and
symmetry for a class of systems with natural growth in the gradient.
