  The symmetry and resonance properties of the Fermi Pasta Ulam chain with
periodic boundary conditions are exploited to construct a near-identity
transformation bringing this Hamiltonian system into a particularly simple
form. This `Birkhoff-Gustavson normal form' retains the symmetries of the
original system and we show that in most cases this allows us to view the
periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville
integrable Hamiltonian. According to the KAM theorem this proves the existence
of many invariant tori on which motion is quasiperiodic. Experiments confirm
this qualitative behaviour. We note that one can not expect it in lower-order
resonant Hamiltonian systems. So the FPU chain is an exception and its special
features are caused by a combination of special resonances and symmetries.
