Examples of reflective projective billiards and outer ghost billiards

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 $4$.