  A wide class of Hamiltonian systems with N degrees of freedom and endowed
with, at least, (N-2) functionally independent integrals of motion in
involution is constructed by making use of the two-photon Lie-Poisson
coalgebra. The set of (N-2) constants of the motion is shown to be a universal
one for all these Hamiltonians, irrespectively of the dependence of the latter
on several arbitrary functions and N free parameters. Within this large class
of quasi-integrable N-dimensional Hamiltonians, new families of completely
integrable systems are identified by finding explicitly a new independent
integral through the analysis of the sub-coalgebra structure of the two-photon
algebra. In particular, new completely integrable N-dimensional Hamiltonians
describing natural systems, geodesic flows and static electromagnetic
Hamiltonians are presented.
