We show the non-integrability of the three-parameter
Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory,
with the exception of the three already known integrable cases. We use
Poincar\'e sections to illustrate the breakdown of regular motion for some
parameter values.Comment: Accepted for publication in SIAM Journal on Applied Dynamical
Systems. Adapted version for arxiv with 19 pages and 11 figure
