Analytic-non-integrability of an integrable analytic Hamiltonian system (Q558080)

The present paper deals with the following question: whether there exists analytic Hamiltonian systems that are \(C^\infty\)-integrable but not analytically integrable. The authors give a positive answer to this question in the case, when the Hamiltonian has a form \[ H(q_1,q_2,p_1,p_2):= (q^2_2+ (q^2_1+ q^2_2)^2) p_1- q_1 q_2 p_2. \] To this end, the authors use power series expansions.