On the integrability of a system describing the stationary solutions in Bose-Fermi mixtures

We study the integrability of a Hamiltonian system describing the stationary solutions in Bose-Fermi mixtures in one dimensional optical lattices. We prove that the system is integrable in the Liouville sense only when it is separable in three generic cases. The proof is based on the differential Galois approach and the Ziglin-Morales-Ramis method.