1:2:2 resonance was studied by Van der Aa and Verhulst who showed its asymptotic integrability, i.e. if H = H_2 + H_3 + … is in Birkhoff normal form such that H_2 is in 1:2:2 resonance, then H_2 + H_3 is integrable. They also generalized the result to the n degrees of freedom 1:2:2:…:2 resonance case.

The original proof of Van der Aaa of asymptotic integrability, using real symplectic coordinates, is quite lengthy. Here I want to write down a simpler proof.

Instead of using real symplectic coordinates and write , we will use complex symplectic coordinates and write

(The problem of finding formal or polynomial first integrals is a purely algebraic problem, so if one can do it over C one can also do it over R).

Then the only monomial functions of degree 3 which Poisson-commute with are . Thus one can write

where are constants.

The quadratic Hamiltonian admits a natural gl(2,C) symmetry: On the 4-dimensional supspace spanned by , the Hamiltonian vector field of can be viewed as the cotangent lifting of the radial vector field on the plane , and thus it commutes with the cotangent lifting of any linear vectore field on the plane . With this symmetry, one can further normalize . This symmetry also implies that all the quadratic functions are first integrals of

As a consequence, the quadratic function

commutes with both and , and so this function is an additional first integral (of degree 2) for the system.

(If, say then in the above formula for can take any other non-zero coefficients ).

The 1:2:…:2 case is similar. One can write as

Then has the form:

where are constants. This resonance admits a natural symmetry, from which one can find additional quadratic commuting first integrals for . In fact, one doesn’t need the whole gl(n-1) algebra, but only some special elements :

Denote by . One can find independent (n-1)-dimensional vectors and independent vectors such that

for any

Then define

where and

Then these functions are additional quadratic commuting first integrals for

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