Non-integrability of the collinear three-body problem (Q536134)

The author considers a collinear three-body problem with particles located on a common line. The mass parameters of particles are restricted to satisfy the ``allowable'' conditions (two branches of the unstable manifold of equilibrium points go up to the separate arms). The non-integrability is studied by applying the Ziglin's analysis. The author proves that the collinear three-body problem with ``allowing'' masses is nonintegrable. Namely, there are no analytical first integrals independent of the total energy and total momentum. The author also studies the triple collision of particles.