Here we study the solutions of any sign of the system −Δu1=∇u2 p,−Δu2=∇u1 q,in a domain of RN,N≧3 and p,q>0,pq>1. We show their relation with Lane–Emden Hardy–Hénon equations −Δp Nw=ɛrσwq,ɛ=±1,where u↦Δp Nu(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–Hénon equations and give nonradial a priori estimates and Liouville type results for the system.