$\omega$-Stable Trigonometries on a Projective Plane

Using the well-known Hrushovski construction, we prove that, for every countable group $G$, there exists an $\omega$-stable trigonometry of the group $G\ast F_\omega$, where $F_\omega$ is the free group of countable rank, on a non-Desarguesian projective plane. We also suggest a new approach to constructing generic models.