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We provide an alternative method for obtaining of compatible Poissonstructures on Lie groups by means of the adjoint representations of Liealgebras. In this way, we calculate some compatible Poisson structures on fourdimensional and nilpotent six dimensional symplectic real Lie groups. Thenusing Magri-Morosi's theorem we obtain new bi-Hamiltonian systems with fourdimensional and nilpotent six dimensional symplectic real Lie groups as phasespaces.

Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups