  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 complete 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 nontrivial set of third-order
Calogero-Moser-like nilpotent integrable equations is obtained.
