Nilpotent integrability, reduction of dynamical systems and a third-order Calogero–Moser system

  • A. Ibort
  • G. Marmo
  • M. A. RodríguezEmail author
  • P. Tempesta


We present an algebraic formulation of the notion of integrability of dynamical systems, based on a nilpotency property of its flow: It can be explicitly described as a polynomial on its evolution parameter. Such a property is established in a purely geometric–algebraic language, in terms both of the algebra of all higher-order constants of the motion (named the nilpotent algebra of the dynamics) and of a maximal Abelian algebra of symmetries (called a Cartan subalgebra of the dynamics). It is shown that this notion of integrability amounts to the annihilator of the nilpotent algebra being contained in a Cartan subalgebra of the dynamics. Systems exhibiting this property will be said to be nilpotent-integrable. Our notion of nilpotent integrability offers a new insight into the intrinsic dynamical properties of a system, which is independent of any auxiliary geometric structure defined on its phase space. At the same time, it extends in a natural way the classical concept of integrability for Hamiltonian systems. An algebraic reduction procedure valid for nilpotent-integrable systems, generalizing the well-known reduction procedures for symplectic and/or Poisson systems on appropriate quotient spaces, is also discussed. In particular, it is shown that a large class of nilpotent-integrable systems can be obtained by reduction of higher-order free systems. The case of the third-order free system is analyzed and a non-trivial set of third-order Calogero–Moser-like nilpotent-integrable equations is obtained.


Dynamical systems Integrable systems Reduction methods Lie algebras 

Mathematics Subject Classification

37N05 37K10 



The authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in RD (SEV-2015/0554). AI would like to thank partial support provided by the MINECO research project MTM2017-84098-P and QUITEMAD+, S2013/ICE-2801. GM would like to acknowledge the support provided by the ‘Cátedras de Excelencia’ Santander/UC3M 2016-17 program. MAR and PT would like to thank partial financial support from MINECO research project FIS2015-63966-P.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Depto. de MatemáticasUniv. Carlos III de MadridLeganésSpain
  2. 2.Instituto de Ciencias Matemáticas, ICMATMadridSpain
  3. 3.Dipartimento di Fisica dell’Universitá “Federico II” di Napoli, Sezione INFN di NapoliComplesso Universitario di Monte S. AngeloNaplesItaly
  4. 4.Depto. de Física TeóricaUniv. Complutense de MadridMadridSpain

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