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We present an algebraic formulation of the notion of integrability ofdynamical systems, based on a nilpotency property of its flow: it can beexplicitly described as a polynomial on its evolution parameter. Such aproperty is established in a purely geometric--algebraic language, in termsboth of the algebra of all higher-order constants of the motion (named thenilpotent algebra of the dynamics), and of a maximal Abelian algebra ofsymmetries (called a Cartan subalgebra of the dynamics). It is shown that thisnotion of integrability amounts to the annihilator of the nilpotent algebrabeing contained in a Cartan subalgebra of the dynamics. Systems exhibiting thisproperty will be said to be nilpotent integrable.  Our notion of nilpotent integrability offers a new insight into the intrinsicdynamical properties of a system, which is independent of any auxiliarygeometric structure defined on its phase space. At the same time, it extends ina natural way the classical concept of complete integrability for Hamiltoniansystems. An algebraic reduction procedure valid for nilpotent integrablesystems, generalizing the well-known reduction procedures for symplectic and/orPoisson systems on appropriate quotient spaces, is also discussed. Inparticular, it is shown that a large class of nilpotent integrable systems canbe obtained by reduction of higher-order free systems. The case of thethird-order free system is analyzed and a nontrivial set of third-orderCalogero-Moser-like nilpotent integrable equations is obtained.

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