Involutive Functionals, Infinite Dimensional Tori, and Neighboring Tori

In the class of smooth periodic functions, we consider Hamiltonian equationsut=K(u), constants of the motionFm(u) that are in involution, and prove an infinite-dimensional version of Liouvilleʹs Theorem. We explain in what sense the generic level set of the functionalsFm(u) is an infinite-dimensional torus, why the solution of the Hamiltonian equation is almost periodic in time, and describe how neighboring generic infinite-dimensional tori are related. The proof of the Theorem is independent of the method of inverse spectral theory and the viewpoint of algebraic curves.