We consider the ground state solutions of the Lane-Emden system with Hénon-type weights − u = |x| β |v| q−1 v, − v = |x| α |u| p−1 u in the unit ball B of R N with Dirichlet boundary conditions, where N 1, α, β 0, p, q > 0 and 1/(p + 1) + 1/(q + 1) > (N − 2)/N. We show that such ground state solutions u, v always have definite sign in B and exhibit a foliated Schwarz symmetry with respect to a unit vector of R N . We also give precise conditions on the parameters α, β, p and q under which the ground state solutions are not radially symmetric.