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 $(L_2-\lambda~L_1)U=0$ is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter $\lambda\in\mathbb{C}$. To construct the operator $L_1$, 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 $L_1\widetilde U=L_2U$ defines a Bäcklund transformation mapping a solution $U$ of the linearized equation to another solution $\widetilde U$ of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.