Integrable Systems and their Recursion Operators

In this paper we discuss the structure of recursion operators. We show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries. This enables us to compute recursion operators more systematically. Under certain conditions (which hold for all examples known to us) Nijenhuis operators are well defined, i.e., they give rise to hierarchies of in-finitely many commuting symmetries of the operator. Moreover, the nonlocal part of a Nijenhuis operator contains the candidates of roots and coroots