  In the series of papers [1-4], the inverse problem of the Birkhoff-Gustavson
normalization was posed and studied. To solve the inverse problem, the
symbolic-computing program named ANFER (Algorithm of Normal Form Expansion and
Restoration) is written up, with which a new aspect of the Bertrand and Darboux
integrability condition is found \cite{Uwano2000}. In this paper, the procedure
in ANFER is presented in mathematical terminology, which is organized on the
basis of the composition of canonical transformations.
