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

Examples of reflective projective billiards and outer ghost billiards