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

We consider the recursion operator approach to the soliton equations related to the generalized Zakharov–Shabat system on the algebra $\mathrm{sl}(n,\mathbb C)$ 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.