Boussinesq hierarchy and bi-Hamiltonian geometry

We study the Boussinesq hierarchy in the geometric context of the theory of bi-Hamiltonian manifolds. First, we recall how its bi-Hamiltonian structure can be obtained by means of a process called bi-Hamiltonian reduction, choosing a specific symplectic leaf of one of the two Poisson structures. Then, we introduce the notion of a bi-Hamiltonian -hierarchy, that is, a bi-Hamiltonian hierarchy that is defined only at the points of the symplectic leaf , and we show that the Boussinesq hierarchy can be interpreted as the reduction of a bi-Hamiltonian -hierarchy.