Analytic integrability of Hamiltonian systems with exceptional potentials

Jaume Llibre, Claudia Valls

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© 2015 Elsevier B.V. All rights reserved. We study the existence of analytic first integrals of the complex Hamiltonian systems of the formH=12 i=12pi2+<sup>Vl</sup>(<sup>q1</sup>,<sup>q2</sup>) with the homogeneous polynomial potential<sup>Vl</sup>(<sup>q1</sup>,<sup>q2</sup>)=α(<sup>q2</sup>-i<sup>q1</sup>)l(<sup>q2</sup>+i<sup>q1</sup>)k-l,l=0,...,k,αC-{0} of degree k called exceptional potentials. In Remark 2.1 of Ref. [7] the authors state: The exceptional potentials <sup>V0</sup>, <sup>V1</sup>, Vk-<inf>1</inf>, <sup>Vk</sup> and Vk/<inf>2</inf> 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 <sup>V0</sup>, <sup>V1</sup>, Vk-<inf>1</inf>, <sup>Vk</sup> and Vk/<inf>2</inf>, have no independent analytic first integral different from the Hamiltonian one.
Original languageEnglish
Pages (from-to)2295-2299
JournalPhysics Letters, Section A: General, Atomic and Solid State Physics
Issue number38
Publication statusPublished - 22 Aug 2015


  • Exceptional potentials
  • Hamiltonian system with 2 degrees of freedom
  • Homogeneous potentials of degree k
  • Integrability


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