Generalized completely integrable systems

Dynamical systems more general than Hamiltonian systems are considered. The role of the Hamiltonian function is played by a $1$-form (not necessarily closed) on a symplectic phase space. A bracket of such forms is introduced and a generalized Liouville theorem on the complete integrability is formulated. This generalization allows us to better understand the meaning of the conditions of the classical theorem on the complete integrability of the Hamilton equations and to reveal the role of tensor invariants.