General Nth-order differential spectral problem: General structure of the integrable equations,nonuniqueness of recursion operator and gauge invariance

The general scalar Gelfand-Dikij-Zakharov-Shabat spectral problem of arbitrary order is considered within the framework of the AKNS method. The general form of the integrable equations is found. Uncertainties which appear in the construction of recursion operator and transformation properties of the integrable equations under the gauge transformations are considered. The manifestly gauge-invariant formulation of the integrable equations is given. It is shown that the intergrable equations under consideration are Hamiltonian ones with respect to the infinite family of Hamiltonian structures.