  The billiard flow in a planar domain acts on its tangent bundle as geodesic
flow with reflections from the boundary. Its trivial first integral is the
squared velocity. Bolotin's Conjecture, now a joint theorem of Bialy, Mironov
and the author, deals with those planar billiards whose flow admits an integral
polynomial in the velocity whose restriction to the unit tangent bundle is
non-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 is
piecewise $C^2$-smooth and contains a nonlinear arc, then it consists of arcs
of conics from a confocal pencil and segments of "admissible lines" for the
pencil; 3) the minimal degree of the integral is either 2, or 4. In 1997 Sergei
Tabachnikov introduced projective billiards: planar curves equipped with a
transversal line field, defining reflection of oriented lines and the
projective billiard flow. They are common generalization of billiards on
constant curvature surfaces, but in general may have no canonical integral. In
a previous paper the author classified those $C^4$-smooth connected nonlinear
planar projective billiards whose flow admits a non-constant integral that is a
rational $0$-homogeneous function of the velocity (with coefficients depending
on the position): they are called rationally $0$-homogeneously integrable. It
was shown that: 1) the underlying curve is a conic; 2) the minimal degree of
integral is equal to two, if the billiard is defined by a dual pencil of
conics; 3) otherwise it can be arbitrary even number. In the present paper we
classify piecewise $C^4$-smooth rationally $0$-homogeneously integrable
projective billiards. Unexpectedly, we show that such a billiard associated to
a dual pencil of conics may have integral of minimal degree 2, 4, or 12. For
the proof of main results we prove dual results for the so-called dual
multibilliards.
