A note on the integrability of Hamiltonian 1 : 2 : 2 resonance

We study the integrability of the Hamiltonian normal form of 1 : 2 : 2 resonance. It is known that this normal form truncated to order three is integrable. The truncated to order four normal form contains too many parameters. For a generic choice of parameters in the normal form up to order four we prove a non-integrability result using Morales-Ramis theory. We also isolate a non-trivial case of integrability.