We consider the elliptic system Delta u = u(p)v(q), Delta v = u(r)v(s) in Omega with the boundary conditions partial derivative u/partial derivative eta = lambda u, partial derivative v/partial derivative eta = mu v on partial derivative Omega, where Omega is a smooth bounded domain of R-N, p, s > 1, q, r > 0, lambda, mu > 0 and eta stands for the outward unit normal. Assuming the "criticality" hypothesis (p - 1)(s - 1)= qr, we completely analyze the values of lambda, mu for which there exist positive solutions and give a detailed description of the set of solutions. Keywords Author Keywords:Elliptic systeminfinitely many solutionsHarnack inequalityRegular 2015FONDECYTFONDECYT