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Over the course of studying billiard dynamics, several questions were raised.One of the questions was, which surfaces satisfy the following property (whichis called Veech's dichotomy): Any direction is either completely periodic oruniquely ergodic. In an important paper Veech gave a sufficient condition forthis dichotomy. He showed that if the stabilizer of a translation surface is alattice in $SL_2(\R)$, then the surface satisfies Veech's dichotomy. Later,Smillie and Weiss proved that this condition is not necessary. They constructeda translation surface which satisfies Veech's dichotomy but is not a latticesurface. Their construction was based on previous work of Hubert and Schmidt,by taking a branched cover over a lattice surface, where the branch locus is asingle non-periodic connection point. In this work we tried to answer thefollowing question: Is there a flat structure obtained from a billiard tablethat satisfies Veech's dichotomy, but its Veech group is not a lattice? Weprove that in the entire list of possible candidates for such a construction,an example does not exist.

Looking for a Billiard Table which is not a Lattice Polygon but Satisfies Veech's Dichotomy