Analytic integrability of Hamiltonian systems with exceptional potentials

• We study the analytic integrability of Hamiltonians 12∑i=12pi2 plus a homogeneous polynomial potential.
• The potentials are α(q2−iq1)l(q2+iq1)k−lα(q2−iq1)l(q2+iq1)k−l, l=0,…,k,α∈C∖{0}l=0,…,k,α∈C∖{0} of degree k.
• As expected, for k   even we prove that the only ones that are completely analytically integrable are the ones with l=0,1,k−1,kl=0,1,k−1,k.

We study the existence of analytic first integrals of the complex Hamiltonian systems of the formH=12∑i=12pi2+Vl(q1,q2) with the homogeneous polynomial potentialVl(q1,q2)=α(q2−iq1)l(q2+iq1)k−l,l=0,…,k,α∈C∖{0} of degree k called exceptional potentials. In Remark 2.1 of Ref. [7] the authors state: The exceptional potentials  V0V0,  V1V1,  Vk−1Vk−1,  VkVkand  Vk/2Vk/2when 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 V0V0, V1V1, Vk−1Vk−1, VkVk and Vk/2Vk/2, have no independent analytic first integral different from the Hamiltonian one.