Geodesic
On rotation invariant integrable systems
Izvestiya. Mathematics , Tome 88 (2024) no. 2, pp. 389-409

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The problem of finding the first integrals of the Newton equations in the $n$-dimensional Euclidean space is reduced to that of finding two integrals of motion on the Lie algebra $\mathrm{so}(4)$ which are invariant under $m\geq n-2$ rotation symmetry fields. As an example, we obtain several families of integrable and superintegrable systems with first, second, and fourth-degree integrals of motion in the momenta. The corresponding Hamilton–Jacobi equation does not admit separation variables in any of the known curvilinear orthogonal coordinate systems in the Euclidean space.
Keywords: differential equation, first integral, symmetry field
Mots-clés : quartic invariant. 
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     title = {On rotation invariant integrable systems},
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A. V. Tsiganov. On rotation invariant integrable systems. Izvestiya. Mathematics , Tome 88 (2024) no. 2, pp. 389-409. http://geodesic.mathdoc.fr/item/IM2_2024_88_2_a9/