  We study the bi-Hamiltonian structures for the hierarchy of a 3-component
generalization of the Degasperis-Procesi (3-DP) equation. We show that all
Hamiltonian functionals in the hierarchy are homogenous, and Hamiltonian
functionals of the hierarchy in the negative direction are local. We construct
two different Liouville transformations by construct a reciprocal
transformation. The associated system for the first one is a reduction of a
negative flow in a modified Yajima-Oikawa hierarchy. The associated system for
the second one and the associated system for another 3-component CH type system
under a equivalent transformation are shown to be different reductions of a
negative generalized mKdV equation, besides the Hamiltonian structures of the
3-DP equation under this Liouville transformation are considered. In addition,
we consider a limit for the 3-DP equation.
