  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 the
new transformations to exchange the role of phase spaces and symmetry groups
and obtain the relations between integrals of motions of these integrable
systems. Finally, we give some examples of real four-dimensional Lie bialgebras
of bi-symplectic type.
