There is a natural action of SL$(2,\mathbb{R})$ on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup $U = \big\{\big(\begin{smallmatrix}1 & * \\ 0 & 1\end{smallmatrix}\big)\big\}$. We classify the U-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL$(2,\mathbb{R})$-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.