Non-integrability of the Kepler and the two body problem on the Heisenberg group

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.