  In this article, we present the Lie transformation algorithm for autonomous
Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a
symplectic structure of the general form. Two examples of normalization in the
restricted three-body problem are given to illustrate the application of the
algorithm in perturbation theory. The efficiency of this algorithm for problems
of asymptotic integration in dynamics is discussed for the case where there is
a need to use non-canonical variables in the phase space.
