On the complete integrability of completely integrable systems

The question of complete integrability of evolution equations associated ton×n first order isospectral operators is investigated using the inverse scattering method. It is shown that forn>2, e.g. for the three-wave interaction, additional (nonlinear) pointwise flows are necessary for the assertion of complete integrability. Their existence is demonstrated by constructing action-angle variables. This construction depends on the analysis of a natural 2-form and symplectic foliation for the groupsGL(n) andSU(n).