Analytic-non-integrability of an integrable analytic Hamiltonian system

We introduce the polynomial Hamiltonian H(q(1), q(2), p(1), p(2)) := (q(2)(2) + (q(1)(2) + q(2)(2))(2))p(1) -q(1)q(2)p(2) and we prove that the associated Hamiltonian system is Liouville-C-infinity-integrable, but fails to be real-analytically integrable in any neighbourhood of an equilibrium point. The proof only uses power series expansions, and is elementary. (c) 2005 Elsevier B.V. All rights reserved.