Geodesic
On Killing tensors in three-dimensional Euclidean space
Teoretičeskaâ i matematičeskaâ fizika, Tome 212 (2022) no. 1, pp. 149-164

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We discuss the properties of second-order Killing tensors in three-dimensional Euclidean space that guarantee the existence of a third integral of motion ensuring the Liouville integrability of the corresponding equations of motion. We prove that in addition to the linear Noether and quadratic Stäckel integrals of motion, there are integrable systems with two quadratic integrals of motion and one fourth-order integral of motion in momenta. A generalization to $n$-dimensional case and to deformations of the standard flat metric is proposed.
Keywords: Hamilton–Jacobi equations, separation of variables, Killing tensors. 
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     author = {A. V. Tsiganov},
     title = {On {Killing} tensors in three-dimensional {Euclidean} space},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {149--164},
     publisher = {mathdoc},
     volume = {212},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2022_212_1_a9/}
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A. V. Tsiganov. On Killing tensors in three-dimensional Euclidean space. Teoretičeskaâ i matematičeskaâ fizika, Tome 212 (2022) no. 1, pp. 149-164. http://geodesic.mathdoc.fr/item/TMF_2022_212_1_a9/