  We obtain the full hamiltonian structure for a parametric coupled KdV system.
The coupled system arises from four different real basic lagrangians. The
associated hamiltonian functionals and the corresponding Poisson structures
follow from the geometry of a constrained phase space by using the Dirac
approach for constrained systems. The overall algebraic structure for the
system is given in terms of two pencils of Poisson structures with associated
hamiltonians depending on the parameter of the Poisson pencils. The algebraic
construction we present admits the most general space of observables related to
the coupled system.
