  The aim of this note is to explain the integrability of an integrable
Boltzmann billiard model, previously established by Gallavotti and Jauslin in
arXiv:2008.01955, alternatively via the viewpoint of projective dynamics. The
additional first integral is shown to be related to the energy of a
corresponding system on a hemisphere. We show that this viewpoint applies to
certain other billiard models in the plane defined through Kepler-Coulomb
problems as well. The approach also leads to a family of integrable billiard
models on the sphere defined through the spherical Kepler-Coulomb problem.
