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A method for constructing the Lax pairs for nonlinear integrable models issuggested. First we look for a nonlinear invariant manifold to thelinearization of the given equation. Examples show that such invariant manifolddoes exist and can effectively be found. Actually it is defined by a quadraticform. As a result we get a nonlinear Lax pair consisting of the linearizedequation and the invariant manifold. Our second step consists of finding anappropriate change of the variables to linearize the found nonlinear Lax pair.The desired change of the variables is again defined by a quadratic form. Themethod is illustrated by the well-known KdV equation and the modified Volterrachain. New Lax pairs are found. The formal asymptotic expansions for theireigenfunctions are constructed around the singular values of the spectralparameter. By applying the method of the formal diagonalization to these Laxpairs the infinite series of the local conservation laws are obtained for thecorresponding nonlinear models.

On a method for constructing the Lax pairs for integrable models via quadratic ansatz