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The billiard flow in a planar domain acts on its tangent bundle as geodesicflow with reflections from the boundary. Its trivial first integral is thesquared velocity. Bolotin's Conjecture, now a joint theorem of Bialy, Mironovand the author, deals with those planar billiards whose flow admits an integralpolynomial in the velocity whose restriction to the unit tangent bundle isnon-constant. It states that 1) if the boundary of such a billiard is$C^2$-smooth, nonlinear and connected, then it is a conic; 2) if it ispiecewise $C^2$-smooth and contains a nonlinear arc, then it consists of arcsof conics from a confocal pencil and segments of "admissible lines" for thepencil; 3) the minimal degree of the integral is either 2, or 4. In 1997 SergeiTabachnikov introduced projective billiards: planar curves equipped with atransversal line field, defining reflection of oriented lines and theprojective billiard flow. They are common generalization of billiards onconstant curvature surfaces, but in general may have no canonical integral. Ina previous paper the author classified those $C^4$-smooth connected nonlinearplanar projective billiards whose flow admits a non-constant integral that is arational $0$-homogeneous function of the velocity (with coefficients dependingon the position): they are called rationally $0$-homogeneously integrable. Itwas shown that: 1) the underlying curve is a conic; 2) the minimal degree ofintegral is equal to two, if the billiard is defined by a dual pencil ofconics; 3) otherwise it can be arbitrary even number. In the present paper weclassify piecewise $C^4$-smooth rationally $0$-homogeneously integrableprojective billiards. Unexpectedly, we show that such a billiard associated toa dual pencil of conics may have integral of minimal degree 2, 4, or 12. Forthe proof of main results we prove dual results for the so-called dualmultibilliards.

On rationally integrable planar dual multibilliards and piecewise smooth projective billiards