Billiards and nonholonomic distributions

In this note, billiards with full families of periodic orbits are considered. It is shown that construction of a convex billiard with a “rational” caustic (i.e., carrying only periodic orbits) can be reformulated as a problem of finding a closed curve tangent to a $(N-1)$-dimensional distribution on a $(2N-1)$-dimensional manifold. The properties of this distribution are described as well as some important consequences for the billiards with rational caustics. A very particular application of our construction states that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist.