Analytic integrability of Hamiltonian systems with exceptional potentials

We study the existence of analytic first integrals of the complex Hamiltonian systems of the form H = 1/2 Sigma(2)(i=1) p(i)(2) + V-l(q(1),q(2)) with the homogeneous polynomial potential V-l(q(1), q(2)) = alpha(q(2) - iq(1))(l)(q(2) +iq(1))(k-l), l = 0, ..., k, alpha is an element of C\ {0} of degree k called exceptional potentials. In Remark 2.1 of Ref. [7] the authors state: The exceptional potentials V-0, V-1, Vk-1, V-k and V-k/2 when k is even are integrable with a second polynomial first integral. However nothing is known about the integrability of the remaining exceptional potentials. Here we prove that the exceptional potentials with k even different from V-0, V-1, Vk-1, V-k and V-k/2, have no independent analytic first integral different from the Hamiltonian one. (C) 2015 Elsevier B.V. All rights reserved.