  This work explores a structure of the Deprit perturbation series and its
connection to a Kato resolvent expansion. It extends the formalism previously
developed for the Hamiltonians linearly dependent on perturbation parameter to
a nonlinear case. We construct a canonical intertwining of perturbed and
unperturbed averaging operators. This leads to an explicit expression for the
generator of the Lie-Deprit transform in any perturbation order. Using this
expression, we discuss a regular pattern in the series, non-uniqueness of the
generator and normalised Hamiltonian, and the uniqueness of the Gustavson
integrals. Comparison of the corresponding computational algorithm with
classical perturbation methods demonstrates its competitiveness for
Hamiltonians with a limited number of perturbation terms.
