  There is a natural action of SL(2,R) on the moduli space of translation
surfaces, and this yields an action of the unipotent subgroup $U =
{\begin{pmatrix} 1 & * 0 & 1 \end{pmatrix}}$.
  We classify the U-invariant ergodic measures on certain special submanifolds
of the moduli space. (Each submanifold is the SL(2,R)-orbit of the set of
branched covers of a fixed Veech surface.) For the U-action on these
submanifolds, this is an analogue of Ratner's Theorem on unipotent flows.
  The result yields an asymptotic estimate of the number of periodic
trajectories for billiards in a certain family of non-Veech rational triangles,
namely, the isosceles triangles in which exactly one angle is $2 \pi/n$, with
$n \ge 5$ and $n$ odd.
