An Elliptic Billiard with the Cosine Law of Refraction at Confocal Quadrics

For a billiard in an elliptic domain, we consider several confocal quadrics, which divide this domain into several regions each filled with homogeneous media with various densities. When crossing the quadric separating two media, it is assumed that the trajectory deviates from a straight line according to the modified (cosine) law of refraction. For the system obtained in this way, the constant of motion is obtained.