Let us briefly summarize our main results. We have
described a generalized version of the powerful tool of normal form
theory for Hamiltonian systems. Using this generalized technique, it is
now possible to analyze any Hamiltonian that is given as a power
series in phase space coordinates.
Even if the Hamiltonian is not given in the form of a power series one
can always expand 
into its Taylor series and normalize the truncated
expansion.
Thus a large variety of Hamiltonian systems can be analyzed in a unified
way.
The most important result of a normalization is the derivation of a
formal integral of motion that, in general, is different from (and often
independent of) the already
known integral . That means that one can obtain substantial new
information about the system by normalization.
Convergence of this formal integral of motion
cannot be taken for granted.
Addressing this problem, we have suggested some new methods for analyzing
the convergence properties of the truncated formal integral. We have found
that these quasi-integrals are of physical interest, since their
convergence properties reflect many of the characteristics of the
corresponding Poincaré plots.
It is certainly necessary to carry the analysis of the convergence
of the quasi-integrals further. Many authors
[10,33,29]
have suggested to study the poles of Padé approximations to the
quasi-integrals. The location and the number of poles of these approximants
then allow to gauge the properties of 
.