Here we study the solutions of any sign of the system −∆u 1 = |∇u 2 | p , −∆u 2 = |∇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-Hénon equations −∆ N p w= εr σ w q , ε = ±1, where u → ∆ N p u (p > 1) is the p-Laplacian in dimension N, q > p − 1 and σ ∈ R. This leads us to explore these equations in not often tackled ranges of the parameters N, p, σ. 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.