  For a system of linear partial differential equations (LPDEs) we introduce an
operator equation for auxiliary operators. These operators are used to
construct a kernel of an integral transformation leading the LPDE to the
separation of variables (SoV). The auxiliary operators are found for various
types of the SoV including conventional SoV in the scalar second order LPDE and
the SoV by the functional Bethe anzatz. The operators are shown to relate to
separable variables. This approach is similar to the position-momentum
transformation to action angle coordinates in the classical mechanics. General
statements are illustrated by some examples.
