Non-quadratic additional conserved quantities in Birkhoff normal forms

For resonant Hamiltonian systems in Poincaré–Birkhoff normal form, the quadratic part of the Hamiltonian is a constant of motion. In the resonant case, the normal form is not unique; this corresponds to free parameters in the solution to homological equations. The “standard” prescription in this case is to set these parameters to zero; however, it was remarked already by Dulac that a different prescription could actually produce a simpler normal form. One such prescription was provided in previous work by the present author; here we discuss how—and under which conditions—this can be used to obtain normal forms which admit, besides the quadratic part, (one or a set of) additional constants of motion of higher degree in nested small neighborhoods of the origin. A concrete example with a cubic natural Hamiltonian in 3 DOF is considered.