Poisson structures and integrable systems

A Poisson coalgebra (Fun(N),Δ) 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 H(n)=λn∑i=1npi+μ∑i,k=1npipjcosν(qi−qj).