Cosymmetries and Nijenhuis recursion operators for difference equations

In this paper we discuss the concept of cosymmetries and co-recursion operators for difference equations and present a co-recursion operator for the Viallet equation. We also discover a new type of factorization for the recursion operators of difference equations. This factorization enables us to give an elegant proof that the pseudo-difference operator presented in Mikhailov et al 2011 Theor. Math. Phys. 167 421–43 is a recursion operator for the Viallet equation. Moreover, we show that the operator is Nijenhuis and thus generates infinitely many commuting local symmetries. The recursion operator and its factorization into Hamiltonian and symplectic operators have natural applications to Yamilov's discretization of the Krichever–Novikov equation