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Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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The analog of the Kepler system defined on the Heisenberg group introduced by Montgomery and Shanbrom in [Fields Inst. Commun., Vol. 73, Springer, New York, 2015, 319–342, arXiv:1212.2713] is integrable on the zero level of the Hamiltonian. We show that in all other cases the system is not Liouville integrable due to the lack of additional meromorphic first integrals. We prove that the analog of the two-body problem on the Heisenberg group is not integrable in the Liouville sense.
Keywords: Kepler problem, two-body problem, Heisenberg group, differential Galois group, integrability, sub-Riemannian manifold. 
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     author = {Tomasz Stachowiak and Andrzej J. Maciejewski},
     title = {Non-Integrability of the {Kepler} and the {Two-Body} {Problems} on the {Heisenberg} {Group}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a73/}
}
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Tomasz Stachowiak; Andrzej J. Maciejewski. Non-Integrability of the Kepler and the Two-Body Problems on the Heisenberg Group. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a73/

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