  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.
