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We construct integrable Hamiltonian systems with Lie bialgebras $({\bf g} ,{\bf \tilde{g}})$ of the bi-symplectic type for which the Poisson-Lie groups${\bf G}$ play the role of the phase spaces, and their dual Lie groups ${\bf{\tilde {G}}}$ play the role of the symmetry groups of the systems. We give thenew transformations to exchange the role of phase spaces and symmetry groupsand obtain the relations between integrals of motions of these integrablesystems. Finally, we give some examples of real four-dimensional Lie bialgebrasof bi-symplectic type.

Exchanging role of the phase space and symmetry group of integrable Hamiltonian systems related to Lie bialgebras of bi-symplectic types