  This paper considers the famous Fermi-Pasta-Ulam chain with periodic boundary
conditions and quartic nonlinearities. Due to special resonances and discrete
symmetries, the Birkhoff normal form of this Hamiltonian system is completely
integrable. We study how the level sets of the integrals foliate the phase
space. Our study reveals all the integrable structure in the low energy domain
of the chain. If the number of particles in the chain is even, then this
foliation is singular. The method of singular reduction shows that the system
has invariant pinched tori and monodromy. Monodromy is an obstruction to the
existence of global action-angle variables. The pinched tori are interpreted as
homoclinic and heteroclinic connections between travelling waves. Thus we
discover a new class of solutions which can be described as direction reversing
travelling waves. They remarkably show an interesting interaction of normal
modes in the absense of energy transfer. These solutions can easily be observed
numerically.
