  The analog of the Kepler system defined on the Heisenberg group introduced in
[9] 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.
