Sergei Tabachnikov suggested a generalization of Birkhoff’s Conjecture to projective billiards, which also involves its versions on surfaces with not zero constant curvature: spherical and hyperbolic plane. It is worded in dual terms. In particular, consider a closed planar convex C-curve with a closed curved lamination (where the C-curve is a sheet) on the outer side. For any straight line L tangent to curve C at a point P, consider the involution germ of the straight line L by P swapping its two points of intersection with each individual leaf. Suppose that this last involution is a projective transformation of the line L for any point P.

Tabashnikov’s Conjecture states that then the curve C is an ellipse, and the leaves of the lamination form a brush of conics.

In the presentation, we will demonstrate the rational version: under the additional hypothesis, that the lamination in question admits a first rational integral.

We will demonstrate an analogous result in the case where the curve C is a real or complex curve germ and we have a laminating germ as above, having a first rational integral. In this general case, the C curve is always a conical curve. But the leaves of the lamination can be algebraic curves of higher degree. We will present the complete classification of these slips with projective transformation.

In the presentation, we will demonstrate the rational version: under the additional hypothesis, that the lamination in question admits a first rational integral.

We will demonstrate an analogous result in the case where the curve C is a real or complex curve germ and we have a laminating germ as above, having a first rational integral. In this general case, the C curve is always a conical curve. But the leaves of the lamination can be algebraic curves of higher degree. We will present the complete classification of these slips with projective transformation.