An important class of physical systems has been called superintegrable because they admit more integrals of motion than degrees of freedom. These integrals form interesting nonabelian algebras, usually finitely generated polynomial ones. Historically the best known superintegrable systems were the Coulomb-Kepler system and the harmonic oscillator. Presently, infinite families of such systems are known and have applications in areas ranging from elementary particle physics to semiconductors and aeronautical engineering. Their mathematical applications mainly concern special function theory, and in particular orthogonal polynomials of continuous and discrete variables.
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