Geometric Interpretation of the Recursion Operators for the Generalized Zakharov-Shabat System in Pole Gauge on the Lie Algebra $A_2$

Abstract

We consider the recursion operator approach to the soliton equations related to a generalized Zakharov-Shabat auxiliary linear system in pole gauge on the Lie algebra $A_2=\mathfrak{sl}(3,\mathbb{C})$ and show that the recursion operator can be identified with the dual to a Nijenhuis tensor for a Poisson-Nijenhuis structure on the manifold of potentials.