Ibort, A.Marmo, G.Rodríguez González, Miguel ÁngelTempesta, Piergiulio2023-06-172023-06-172019-100373-311410.1007/s10231-019-00828-xhttps://hdl.handle.net/20.500.14352/13904© 2019 Springer Heidelberg. 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 `Catedras de Excelencia' Santander/UC3M 2016-17 program. MAR and PT would like to thank partial financial support from MINECO research project FIS2015-63966-P.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.engjournal articlehttp://dx.doi.org/10.1007/s10231-019-00828-xhttps://link.springer.com/open access51-73Dynamical systemsIntegrable systemsReduction methodsLie algebras37N0537K10.Física-Modelos matemáticosFísica matemática