On the Darboux integrability of the logarithmic galactic potentials

We study the logarithmic Hamiltonians H=(px2+py2)∕2+log(1+x2+y2∕q2)1∕2, which appear in the study of the galactic dynamics. We characterize all the invariant algebraic hypersurfaces and all exponential factors of the Hamiltonian system with Hamiltonian H. We prove that this Hamiltonian system is completely integrable with Darboux first integrals if and only if q=±1.