Non-Integrability of the Planar Three-Body Problem with Inverse-Square Potential

We consider the planar three-body problem with generalized force
potential u : R_{>0} → R. Some non-integrability results for these
systems have been obtained by analyzing the variational equations
along the homothetic solutions. But we can not apply it to several
exceptional cases. For example, when the system has inverse-square
potentials, i.e. u(r) = r^{-2}, the variational equations along the
homothetic solutions are always solvable. We obtain sufficiently
conditions for non-integrability for these exceptional cases by
focusing on some particular solutions that are different from
homothetic solutions.