  In \cite{hab-2016,hab-2017}, Habibullin \emph{et.al} proposed an approach to
construct Lax pairs of a nonlinear integrable partial differential equation
(PDE), where one is the linearized equation of the studied PDE and the other is
the invariant manifold of the linearized equation. In this paper, we show that
the invariant manifold is the characteristic of a generalized conditional
symmetry of the system composed of the studied PDE and its linearized PDE. Then
we give an upper order bound of the invariant manifold which provides a
theoretical basis for a complete classification of such type of invariant
manifold. Moreover, we suggest a modified method to construct Lax pair of the
KdV equation which can not be obtained by the original method in
\cite{hab-2016,hab-2017}.
