Geometric Theory of the Recursion Operators for the Generalized Zakharov–Shabat System in Pole Gauge on the Algebra sl(n,C) with and without Reductions

Abstract. We consider the recursion operator approach to the soliton equations related to the generalized Zakharov–Shabat system on the algebra sl(n,C) in pole gauge both in the general position and in the presence of reductions. We present the recursion operators and discuss their geometric meaning as conjugate to Nijenhuis tensors for a Poisson–Nijenhuis structure defined on the manifold of potentials. Key words: Lax representation; recursion operators; Nijenhuis tensors 2010 Mathematics Subject Classification: 35Q51; 37K05; 37K1