  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. In the paper we
study its following generalized dual version stated by S.Tabachnikov. Consider
a closed smooth strictly convex curve $\gamma\subset\mathbb{RP}^2$ equipped
with a dual billiard structure: a family of non-trivial projective involutions
acting on its projective tangent lines and fixing the tangency points. Suppose
that its outer neighborhood admits a foliation by closed curves (including
$\gamma$) such that the involution of each tangent line permutes its
intersection points with every leaf. Then $\gamma$ and the leaves are conics
forming a pencil. We prove positive answer in the case, when the curve $\gamma$
is $C^4$-smooth and the foliation admits a rational first integral. To this
end, we show that each $C^4$-smooth germ $\gamma$ of planar curve carrying a
rationally integrable dual billiard structure is a conic and classify all the
rationally integrable dual billiards on (punctured) conic. They include the
dual billiards induced by pencils of conics, two infinite series of exotic dual
billiards and five more exotic ones.
