  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.
