A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L 1 −1 L 2 , where the linear differential operators L 1 and L 2 are chosen such that the ordinary differential equation (L 2 −λL 1 )U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L 2 , we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L 1 U ˜ = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution U ˜ of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.