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A complete classification is presented of quantum and classical superintegrable systems in $E_2$ that allow the separation of variables in polar coordinates and admit an additional integral of motion of order three in the momentum.
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I. Fris, V. Mandrosov, Ya. A. Smorodinsky, M. Uhlir and P. Winternitz, On higher symmetries in quantum mechanics, Phys. Lett. 16
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P. Letourneau and L. Vinet, Superintegrable systems: Polynomial algebras and quasiexactly solvable Hamilotnians, Ann. Phys. (N.Y.) 243
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J. Hietarinta, Pure quantum integrability, Phys. Lett. A 246
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A. V. Tsiganov, The Drach superintegrable systems, J. Phys. A: Math. Theor. 33
2000
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C. M. Cosgrove, Higher order Painlevé equations in the polynomial class I. Bureau symbol P2, Stud. Appl. Math. 104:1-65 (2000)
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C. M. Cosgrove, Chazy Classes IX-XI of third order differential equations, Stud. Appl. Math. 104:104-228 (2000)
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M. B. Sheftel, P. Tempesta et P. Winternitz, Superintegrable systems in quantum mechanics and classical Lie theory, J. Math. Phys. 42
2001
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P. Tempesta, A. V. Turbiner et P. Winternitz, Exact solvability of superintegrable systems, J. Math. Phys. 42
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P. Winternitz, Superintegrability with second and third order integral of motion, Rus. J. Nucl. Phys. 72(5)
2009
Later among the works it cites.
E.G. Kalnins, J. Kress, W. Miller Jr and S. Post, Structure theory for second order 2D superintegrable systems with 1 parameter potential, Sigma 5, 008 (2009)
2009
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F. Tremblay, A. V. Turbiner and P. Winternitz, An infinite family of solvable and integrable quantum systems on a plane, J. Phys. A: Math. Theor. 42
2009
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2009
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I. Marquette, Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials. J. Math. Phys. 50
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S. Gravel and P. Winternitz, Superintegrable systems with third-order integrals in classical and quantum mechanics, J. Math. Phys. 43
2002
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S. Gravel, Hamiltonians separable in cartesian coordinates and third-order integrals of motion, J. Math. Phys. 45
2004
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C. Daskaloyannis and K. Ypsilantis, Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold, J. Math. Phys. 47
2006
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C. M. Cosgrove, Higher order Painlevé equations in the polynomial class II. Bureau symbol P1, Stud. Appl. Math. 116:321-413 (2006)
2006
Cited alongside, same era.
I. Marquette and P. Winternitz, Polynomial Poisson algebras for classical superintegrable systems with a third-order integral of motion, J. Math. Phys. 48
2007
Cited alongside, same era.
I. Marquette and P. Winternitz, Superintegrable systems with third order integrals of motion , J. Phys. A: Math. Theor. 41
2008
Cited alongside, same era.
E.G. Kalnins, W. Miller Jr and S. Post, Models for quadratic algebras associated with second oderder superintegrable systems in 2D, Sigma 4, 008 (2008)
2008
Cited alongside, same era.
2009
Later among the works it cites.
I. Marquette, Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé transcendent potentials, J. Math. Phys. 50
2009
Later among the works it cites.
I. Marquette, Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion, J. Math. Phys. 50
2009
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I. Marquette, Superintegrability and higher order polynomial algebras I, arXiv:0908.4399v1 (2009)
2009
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I. Marquette, Superintegrability and higher order polynomial algebras II, arXiv:0908.4432v1 (2009)
2009
Later among the works it cites.
F. Tremblay, A. V. Turbiner and P. Winternitz, Periodic orbits for an infinite family of classical superintegrable systems, J. Phys. A: Math. Theor. 43
2010
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C. Quesne, Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd k, J. Phys. A: Math. Theor. 43
2010
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E.G. Kalnins, J. M.Kress, W. Miller Jr, Families of classical subgroup separable superintegrable systems, J. Phys. A: Math. Theor. 43
2010
Closest in time.