Title: On a direct algorithm for constructing recursion operators and Lax pairs for integrable models,
Abstract: We suggested an algorithm for searching the recursion operators for nonlinear
integrable equations. It was observed 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 in such a way 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 $\lambda\in \textbf{C}$. For constructing the operator $L_1$ we use
the concept of the invariant manifold which is a generalization of the
symmetry. Then for searching $L_2$ we take an auxiliary linear equation
connected with the linearized equation by the Darboux transformation.
Connection of the invariant manifold with the Lax pairs and the
Dubrovin-Weierstrass equations is discussed.