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This work explores a structure of the Deprit perturbation series and itsconnection to a Kato resolvent expansion. It extends the formalism previouslydeveloped for the Hamiltonians linearly dependent on perturbation parameter toa nonlinear case. We construct a canonical intertwining of perturbed andunperturbed averaging operators. This leads to an explicit expression for thegenerator of the Lie-Deprit transform in any perturbation order. Using thisexpression, we discuss a regular pattern in the series, non-uniqueness of thegenerator and normalised Hamiltonian, and the uniqueness of the Gustavsonintegrals. Comparison of the corresponding computational algorithm withclassical perturbation methods demonstrates its competitiveness forHamiltonians with a limited number of perturbation terms.

Generalisation of the explicit expression for the Deprit generator to Hamiltonians nonlinearly dependent on small parameter