Hierarchies of Symplectic Structures for sl(3,C) Zakharov-Shabat Systems in Canonical and Pole Gauge with Z2×Z2 Reduction of Mikhailov Type

We consider the theory of the hierarchies of nonlinear evolution equations associated with two gauge-equivalent systems we denote by L± and GMV± (GMVsystem). They are obtained from the Generalized Zakharov-Shabat system on sl(3,C) in general position making a Z2 × Z2 reductions of Mikhailov type in canonical and in pole gauge respectively. Using the Recursion Operators approach and expansions over the adjoint solutions we study the symplectic structures of the hierarchies of the nonlinear evolution equations associated with L± and GMV± and calculate the relation between them.