Geodesic
Superintegrable Systems with Third-Order Integrals in Classical and Quantum Mechanics
Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 1, pp. 97-107

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We review systems in $E(2)$ that are separable in Cartesian coordinates and admit a third-order integral both in quantum mechanics and in classical mechanics. Differences and similarities between those two cases are illustrated by numerous examples. Many of these superintegrable systems are new, and a relation is seen between superintegrable potentials and Painlevé transcendents.
Keywords: integrals of motion, superintegrability, separation of variables. 
@article{TMF_2003_137_1_a10,
     author = {S. Gravel},
     title = {Superintegrable {Systems} with {Third-Order} {Integrals} in {Classical} and {Quantum} {Mechanics}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {97--107},
     publisher = {mathdoc},
     volume = {137},
     number = {1},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2003_137_1_a10/}
}
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S. Gravel. Superintegrable Systems with Third-Order Integrals in Classical and Quantum Mechanics. Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 1, pp. 97-107. http://geodesic.mathdoc.fr/item/TMF_2003_137_1_a10/