Abstract We prove regularity results for the solutions of the equation -Δ m u = h(x), such as summability properties of the second derivatives and summability properties of . Analogous results were recently proved by the authors for the equation -Δ m u = f(u). These results allow us to extend to the case of systems of m-Laplace equations, some results recently proved by the authors for the case of a single equation. More precisely we consider the problem and we prove regularity properties of the solutions as well as qualitative properties of the solutions. Moreover we get a geometric characterization of the critical sets Z u ≡ {x ∈ Ω | Du(x) = 0} and Zv ≡ Dv(x) = 0. In particular we prove that in convex and symmetric domains we have Z u = {0} = Z v , assuming that 0 is the center of symmetry.