The concept of integrability of Hamiltonian systems goes back at least to the 19th century. The idea of integrability in classical mechanics was formalised by J. Liouville. A finite-dimensional Hamiltonian system with n degrees of freedom is called “Liouville integrable” or “completely integrable” if it allows n functionally independent integrals of motion that are well defined functions on phase space and are in involution. In classical mechanics the equations of motion for a Liouville integrable system can be, at least in principle, reduced to quadratures. A completely integrable system in quantum mechanics is defined similarly. It should allow n commuting integrals of motion (including the Hamiltonian) that are well defined operators in the enveloping algebra of the Heisenberg algebra, or some generalization of this enveloping algebra. In quantum mechanics complete integrability does not guarantee that the spectral problem for the Schrödinger operator can be solved explicitly, or even that the energy levels can be calculated algebraically.

An n-dimensional integrable Hamiltonian system that admits more than n integrals of motion is called “superintegrable”. Systems with 2n – 1 integrals, with at least one subset of n of them in involution, are “maximally superintegrable”. Such systems, namely the Kepler-Coulomb system and the harmonic oscillator, played a pivotal role in the development of physics and mathematics. Trajectories in classical maximally superintegrable systems can at least in principle be calculated algebraically (without using any calculus). Ironically, calculus was invented in order to calculate orbits in the Kepler system.