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This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. We investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is constant. These structures called affine or modified Lie-Poisson structures are involved in the integrability of certain Euler equations that arise as models of shallow water waves.doi:10.1098/rsta.2007.2012 pmid:17360267 fatcat:hlbgtg4vozhdpm3xfjh7kwlxoa