Poisson structures and integrable systems

A Poisson coalgebra is used to construct integrable Hamiltonian systems. We consider a Poisson structure given by the bivector P=Pab(x)(∂/∂xa) (∂/∂xb), x R3, which does not form a Lie algebra with respect to the Poisson bracket {xa,xb}P=Pab(x). We prove that this coalgebra may be used to generate integrable Hamiltonian systems. As an example we give the Poisson tensor P=νx3(∂/∂x2) (∂/∂x2)+νx2(∂/∂x3) (∂/∂x1)−(ν/2) (∂/∂x2) (∂/∂x3) and we show that it is linked with the Calogero system .