Let $M$ be a compact Riemannian manifold with metric $g$ and associated Riemannian volume $\nu$ and geodesic flow $G_t : UTM \rightarrow UTM$, where the unit tangent bundle is indicated. Let $X_j \subset UTM$ for $1 \le j \le n$ be open disjoint codimension one submanifolds transversal to $G_t$, i.e., local cross sections (a global cross section does not exist).

Is it possible to choose a metric $g'$ on $M$ with geodesic flow $G'_t = G_t$ and $\nu'_1(X_j) \equiv 1$?

NB. Here $\nu'_1$ denotes the induced codimension one [relative] measure on $UTM$.

This question was prompted by a helpful comment to this one.