Integrable models are systems in quantum physics which can be solved exactly because they have a large number of conserved quantities and therefore a high degree of symmetry.
These integrable models appear in many guises: field theories, spin chains, models of statistical mechanics and models of a fixed number of interacting particles. They can also be subsumed into models with even more symmetry, such as conformal field theories and 'superintegrable' models, including the harmonic oscillator, the Kepler system in Newtonian gravity and the hydrogen atom.
The last few decades have seen a resurgence of exactly-solvable physical systems that appear ubiquitously in modern fundamental physics. These have been at the heart of recent advances in our understanding of the relationship between gauge field theory and string theory (gauge-string correspondence).
The mathematical beauty of such models is often seen in new algebraic structures and we have been involved in the discovery and development of many.