by projectors to its energy eigenstates. However, this does not imply any special dynamical structure. To explain quantum integrability, it is helpful to consider the free particle setting. Here all dynamics are one-body reducible. A quantum system is said to be integrable if the dynamics are two-body reducible. The Yang–Baxter equation is a consequence of this reducibility and leads to trace identities which provide an infinite set of conserved quantities. All of these ideas are incorporated into the quantum inverse scattering method where the algebraic Bethe ansatz can be used to obtain explicit solutions. Examples of quantum integrable models are the Lieb–Liniger model, the Hubbard model and several variations on the Heisenberg model. Some other types of quantum integrability are known in explicitly time-dependent quantum problems, such as the driven Tavis-Cummings model. Exactly solvable models In physics, completely integrable systems, especially in the infinite-dimensional setting, are often referred to as exactly solvable models. This obscures the distinction between integrability in the Hamiltonian sense, and the more general dynamical systems sense. There are also exactly solvable models in statistical mechanics, which are more closely related to quantum integrable systems than classical ones. Two closely related methods: the Bethe ansatz approach, in its modern sense, based on the Yang–Baxter equations and the quantum inverse scattering method, provide quantum analogs of the inverse spectral methods. These are equally important in the study of solvable models in statistical mechanics. An imprecise notion of "exact solvability" as meaning: "The solutions can be expressed explicitly in terms of some previously known functions" is also sometimes used, as though this were an intrinsic property of the system itself, rather than the purely calculational feature that we happen to have some "known" functions available, in terms of which the solutions may be expressed. This notion has no intrinsic meaning, since what is meant by "known" functions very often is defined precisely by the fact that they satisfy certain given equations, and the list of such "known functions" is constantly growing. Although such a characterization of "integrability" has no intrinsic validity, it often implies the sort of regularity that is to be expected in integrable systems. List of some well-known integrable systems Classical mechanical systems (finite-dimensional phase space) Harmonic oscillators in n dimensions Central force motion (exact solutions of classical central-force problems) Two center Newtonian gravitational motion Geodesic motion on ellipsoids Neumann oscillator Lagrange, Euler, and Kovalevskaya tops Integrable