  In the class of projective billiards, which contains the usual billiards, we
exhibit counter-examples to Ivrii's conjecture, which states that in any planar
billiard with smooth boundary the set of periodic orbits has zero measure. The
counter-examples are polygons admitting a $2$-parameters family of $n$-periodic
orbits, with $n$ being either $3$ or any even number grower than $4$.
