Symmetry results for stable and monotone solutions to fibered systems of PDEs

We study the symmetry properties for solutions of elliptic systems of the type {ll}-\dive(a_1(x,|\nabla u^1|(X))\nabla u^1(X))=F_{1}(x, u^1(X),..., u^n(X)), ... -\dive(a_n(x,|\nabla u^n|(X))\nabla u^n(X))=F_{n}(x, u^1(X),..., u^n(X)), where $x\in \R^m$ with $1\leq m< N$, $X=(x,y)\in \R^m\times \R^{N-m}$, and $F_{1},..., F_{n}$ are the derivatives with respect to $\xi^1,..., \xi^n$ of some $F=F(x,\xi^1,..., \xi^n)$ such that for any $i=1,..., n$ and any fixed $(x,\xi^1,..., \xi^{i-1},\xi^{i+1},..., \xi^n)\in \R^m\times \R^{n-1}$ the map $\xi^i\to F(x,\xi^1,...,\xi^i,..., \xi^n)$ belongs to $C^2(\R)$. We obtain a Poincar\'e-type formula for the solutions of the system and we use it to prove a symmetry result both for stable and for monotone solutions.