  We study the near threshold behavior of cross sections of low-energy
antiproton scattering off the ground and excited states of positronium with
zero total orbital momentum $L=0$. In our computational experiment, the
existence of singularities called the Gailitis-Damburg oscillations above the
thresholds of excited states of positronium and antihydrogen atoms is
confirmed. In the future the obtained results can be useful for developing
proposals for improving the conditions of experiments with antimatter.
