This paper shows that any symplectic diffeomorphism $\mathbf{F} :(\Sigma^{2n},\eta) \to (\Sigma^{2n},\eta)$ can be embedded as a subsystem of a Liouville-integrable Hamiltonian flow on some symplectic manifold. If F is real-analytic, then the flow can be chosen to be real-analytic, but it is Liouville integrable with smooth first integrals. Examples are constructed of integrable, volume-preserving Hamiltonian flows on Poisson manifolds whose metric entropy with respect to the volume form is positive. Completely integrable Hamiltonian flows on a symplectic manifold are constructed which have positive metric entropy with respect to an invariant probability measure that is absolutely continuous with respect to the canonical volume form.