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We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the
adjoint representations of Lie algebras. In this way we calculate some compatible Poisson structures on four
dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi’s theorem we
obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie
groups as phase spaces.
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