  Over the course of studying billiard dynamics, several questions were raised.
One of the questions was, which surfaces satisfy the following property (which
is called Veech's dichotomy): Any direction is either completely periodic or
uniquely ergodic. In an important paper Veech gave a sufficient condition for
this dichotomy. He showed that if the stabilizer of a translation surface is a
lattice in $SL_2(\R)$, then the surface satisfies Veech's dichotomy. Later,
Smillie and Weiss proved that this condition is not necessary. They constructed
a translation surface which satisfies Veech's dichotomy but is not a lattice
surface. 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 a
single non-periodic connection point. In this work we tried to answer the
following question: Is there a flat structure obtained from a billiard table
that satisfies Veech's dichotomy, but its Veech group is not a lattice? We
prove that in the entire list of possible candidates for such a construction,
an example does not exist.
