On classification of polynomial Hamiltonians with non-degenerated linear-stable singular point

We study the classification of polynomial Hamiltonians with the non-degenerated linear-stable singular point on the two-dimensional complex plane with respect to the action of the group of polynomial symplectic automorphisms. For each Hamiltonian one can associate the set of polynomials in three variables and the finite group. These variables are the components of the Birkhoff normal form of our Hamiltonian, and this group is the Galois group of the finite-dimensional extension of the fields, which is generated by our polynomials. Using these objects we provide the equivalence criterion for two polynomial Hamiltonians.