An integrable billiard system in homogeneous media separated by confocal quadrics

The talk will focus on aa apparently new integrable system.
Imagine that two media of different optical densities are located in the area bounded by an ellipse. The media are separated by a confocal quadric. A light beam at the confocal quadric is either refracted according to Snellius' law, or experiences a complete internal reflection. Experimentally, it is not difficult to verify that such a system is not integrable. If the law of refraction is modified: the sinuses of the incident and refracted rays are replaced by cosines, then such a system will have the first integral integral.
It is very intriguing that in some cases this first integral takes values on a circle, i.e. its value is determined up to the addition of some well-defined constant. Explicit formulas will be presented for such first integrals in different situations, the surfaces of the constant value of the first integral and their bifurcations in the vicinity of the critical value of the first integral are explicitly described.