Analytic-non-integrability of an integrable analytic Hamiltonian system

We introduce the polynomial Hamiltonian H(q1,q2,p1,p2):=(q22+(q12+q22)2)p1−q1q2p2 and we prove that the associated Hamiltonian system is Liouville-C∞-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.