theorem; i.e., the existence of action-angle variables. General dynamical systems have no such conserved quantities; in the case of autonomous Hamiltonian systems, the energy is generally the only one, and on the energy level sets, the flows are typically chaotic. A key ingredient in characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable distribution) if, locally, it has a foliation by maximal integral manifolds. But integrability, in the sense of dynamical systems, is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds. Integrable systems do not necessarily have solutions that can be expressed in closed form or in terms of special functions; in the present sense, integrability is a property of the geometry or topology of the system's solutions in phase space. General dynamical systems In the context of differentiable dynamical systems, the notion of integrability refers to the existence of invariant, regular foliations; i.e., ones whose leaves are embedded submanifolds of the smallest possible dimension that are invariant under the flow. There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation. This concept has a refinement in the case of Hamiltonian systems, known as complete integrability in the sense of Liouville (see below), which is what is most frequently referred to in this context. An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describe evolution equations that either are systems of differential equations or finite difference equations. The distinction between integrable and nonintegrable dynamical systems has the qualitative implication of regular motion vs. chaotic motion and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated into an exact form. Hamiltonian systems and Liouville integrability In the special setting of Hamiltonian systems, we have the notion of integrability in the Liouville sense. (See the Liouville–Arnold theorem.) Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated with the invariants of the foliation span the tangent distribution. Another way to state this is that there exists a maximal set of Poisson commuting invariants (i.e., functions on the phase space whose