On rationally integrable planar dual multibilliards and projective billiards

A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then the billiard is an ellipse. It was studied by many mathematicians, including H.Poritsky, M.Bialy, S.Bolotin, A.Mironov, V.Kaloshin, A.Sorrentino and others.
We present positive results on its dual version stated by S.Tabachnikov, for so-called dual billiards (curves equipped with a family of projective involutions acting on tangent lines and fixing tangency points), under the condition of existence of a rational function whose restriction to each tangent line is invariant under its involution. We will discuss a related open problem on classification of polygonal projective billiards (polygons equipped with a transversal line field on the boundary, defining reflection of lines from the boundary) with rationally integrable billiard flow. It is closely related to a problem on actions of groups of Cremona transformations generated by de Jonquieres involutions.