dg.differential geometry - Can the geodesic flow be preserved by an inhomogeneous rescaling of a cross section?

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.