A 3X3 MATRIX SPECTRAL PROBLEM FOR AKNS HIERARCHY AND ITS BINARY NONLINEARIZATION

A 3 x 3 matrix spectral problem for AKNS soliton hierarchy is introduc ed and the corresponding Bargmann symmetry constraint involving Lax pa irs and adjoint Lax pairs is discussed. An explicit new Poisson algebr a is proposed and thus the Liouville integrability is established for the nonlinearized spatial system and a hierarchy of nonlinearized temp oral systems under the control of the nonlinearized spatial system. Th e obtained nonlinearized Lax systems, in which the nonlinearized spati al system is intimately related to stationary AKNS flows, lead to a so rt of new involutive solutions to each AKNS soliton equation. Therefor e, the binary nonlinearization theory is successfully extended to a ca se of 3 x 3 matrix spectral problem for AKNS hierarchy.