  In this paper we shall study qualitative properties of a $p$-Stokes type
system, namely $$ -{\boldsymbol \Delta}_p{\boldsymbol u}=-\operatorname{\bf
div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x,{\boldsymbol
u})\,\, \mbox{ in $\Omega$},$$ where ${\boldsymbol \Delta}_p$ is the
$p$-Laplacian vectorial operator. More precisely, under suitable assumptions on
the domain $\Omega$ and the function $\boldsymbol{ f}$, it is deduced that
system solutions are symmetric and monotone. Our main results are derived from
a vectorial version of the weak and strong comparison principles, which enable
to proceed with the moving-planes technique for systems. As far as we know,
these are the first qualitative kind results involving vectorial operators.
