Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom

The normal forms of the Hamiltonian 1:2:ω resonances to degree three for ω = 1, 3, 4 are studied for integrability. We prove that these systems are non-integrable except for the discrete values of the parameters which are well known. We use the Ziglin–Morales–Ramis method based on the differential Galois theory.