Integrable equations connected with the Poisson algebra

The general $r$-matrix construction of integrable Hamiltonian systems is applied to Poisson algebras which are function algebras on symplectic manifolds with commutator given by the Poisson bracket. Two types of integrable systems are described: Hamiltonian systems on the group of symplectic diffeomorphisms whose Hamiltonians are sums of a left-invariant kinetic energy and a potential, and systems of two first order equations for two functions of one variable.