Abstract.We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.

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A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold of the linearization of the given equation. Examples show that such an invariant manifold does exist and can effectively be found. Actually, it is defined by a quadratic form. As a result we get a nonlinear Lax pair consisting of the linearized equation and the invariant manifold. Our second step consists of finding an appropriate change of the variables to linearize the found nonlinear Lax pair. The desired change of the variables is again defined by a quadratic form. The method is illustrated by the well-known KdV equation, the modified Volterra chain and a less studied coupled lattice connected to the affine Lie algebra . New Lax pairs are found. The formal asymptotic expansions for their eigenfunctions are constructed around the singular values of the spectral parameter. By applying the method of the formal diagonalization to these Lax pairs, the infinite series of the local conservation laws are obtained for the corresponding nonlinear models.

A notion of the generalized invariant manifold for a nonlinear integrable lattice is considered. Earlier it has been observed that this kind objects provide an effective tool for evaluating the recursion operators and Lax pairs. In this article we show with an example of the Volterra chain that the generalized invariant manifold can be used for constructing exact particular solutions as well. To this end we first find an invariant manifold depending on two constant parameters. Then we assume that ordinary difference equation defining the generalized invariant manifold has a solution polynomially depending on one of the spectral parameters and derive ordinary difference and differential equations, for the roots of the polynomials. In the simplest nontrivial case we constructed exact common solutions to these equations. As a result we derived for the Volterra chain a new explicit solution, displaying unusual behavior.

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 − λL 1 )U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ 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.