 Solution

Let $\pi :SM\to M$ be the unit tangent bundle projection. Write $X$ for the geodesic flow generator and $V$ for the vertical vector field of a surface unit tangent bundle, positively rotating $v$ toward $iv$. The acceleration prescribed in the question changes only this velocity angle, so the surface thermostat generator isIt never vanishes: $d\pi (F)=v\ne 0$. Also $F$ and $V$ are linearly independent, since $d\pi (V)=0$.

The vertical Maslov cycle of a surface thermostat isThere is exactly one such plane in each fibre, so ${\Lambda }_V$ is the image of a smooth section. A two-plane containing $F$ corresponds to a line in the rank-two quotient $T(SM)/\mathbb{R}F$; thus $\Lambda (SM)$ is a projective bundle with fibre ${\mathbb{RP}}^1$. Its total dimension is four and the cycle has dimension three, making it a smooth hypersurface. In this two-dimensional transverse space the Maslov cycle means the projective line meeting the distinguished vertical line, which here is just that distinguished line itself in each fibre.