Action-angle variables for a multicomponent nonlinear Schrödinger equation

An auxiliary linear problem for the multicomponent nonlinear Schrödinger (NS) equation is the Dirac matrix system. For this system we give basic formulas for the direct and inverse problems of scattering theory. It is shown how the reduction reduces the invariance group of NS equation. The Riccati matrix equation leads to recurrent relations for the local densities preserving the motion integrals and allows us to transfer the relations defining the reduction to the data of the scattering. The action-angle variables are selected from the elements of the transition matrix after a special transformation from the invariance group and factorizations.