Birkhoff normal forms are special normal forms of Hamiltonian systems near equilibria and the problem of determining whether they are non-integrable has attracted much attention in the field of dynamical systems. In this talk we prove that three-degree-of-freedom Birkhoff normal forms are non-integrable when they are in 1:2:ω resonance for ω =1,2 or 4. Recently, Christov  also studied non-integrability of the same normal forms although his proofs contained some errors.
 O. Christov, Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom, Celest. Mech. Dyn. Astr., 112 (2012), 149-167.