 First-order canonical removal of a cubic oscillator perturbation

The accompanying momentum map is $p=P+2\lambda XP/(m{\omega }^2)$. Its Poisson bracket is one up to second-order corrections. Substitution into $p^2/(2m)+m{\omega }^2{x}^2/2+\lambda x^3$ cancels all terms linear in $\lambda$, leaving a harmonic Hamiltonian to that order. The map is local and perturbative, not an exact global removal of an unbounded cubic potential.