Non-Canonical Symmetries, Bi-Hamiltonian Structures, and Complete Integrability

Abstract

It is shown that a non-canonical symmetry of a finite-dimensional Hamiltonian system leads to a bi-Hamiltonian structure for the system. If the recursion operator has a vanishing Nijenhuis tensor and minimal degeneracy, it generates a sequence of conserved quantities in involution. The recursion operator is also the Lax matrix of an iso spectral representation, and its eigenvalues are conserved quantities in involution. If these conserved quantities are functionally independent, the system is completely integrable.