Projectively Equivalent Metrics, Billiards, and Nijenhuis Geometries

We will discuss connections between two seemingly completely different areas of mathematics: the theory of projectively equivalent metrics and integrable billiards in domains bounded by quadrics.
The first stage is devoted to the classification of geodesically consistent pairs $g$, $L$ in dimension two. Such pairs prove to be extremely interesting objects. In particular, the level lines of the eigenvalues of the operator $L$ define a family of quadrics with remarkable focal properties (in the Euclidean case, this is precisely an elliptic coordinate system, i.e., a family of confocal quadrics).
Furthermore, such pairs $g$, $L$ are associated with an integrable system whose additional integral is preserved under the reflection of a material point from a curve of such a family (reflection should be considered in the sense of the corresponding flat metric, i.e., pseudo-Euclidean reflections and Minkowski billiards arise). This allows us to define a large class of billiards, a significant portion of which are new.