  Here we study the solutions of any sign of the system --$\Delta$u 1 =
|$\nabla$u 2 | p , --$\Delta$u 2 = |$\nabla$u 1 | q , in a domain of R N , N 3
and p, q > 0, pq > 1.. We show their relation with Lane-Emden Hardy-H{\'e}non
equations --$\Delta$ N p w= $\epsilon$r $\sigma$ w q , $\epsilon$ = $\pm$1,
where u $\rightarrow$ $\Delta$ N p u (p > 1) is the p-Laplacian in dimension N,
q > p -- 1 and $\sigma$ $\in$ R. This leads us to explore these equations in
not often tackled ranges of the parameters N, p, $\sigma$. We make a complete
description of the radial solutions of the system and of the Hardy-Henon
equations and give nonradial a priori estimates and Liouville type results.
