I wanted to know if there is a sort of "constrained Moser Trick". Suppose we have a planar grap $G \subseteq \mathbb{R}^2$, with $(0,0)$ as a vertex. Suppose to have some volume form $\omega = g_\ast(dx \wedge dy)$, where $g$ is a diffeomorphism of the plane that send $0$ into $0$.

By construction, the Moser equation

$\frac{d}{dt}\omega_t + \mathcal{L}_{X_t}\omega_t = 0$ with $\omega_t = \omega(1-t) + tdx\wedge dy$

can be solved, and we can find some $h$ (in this case $= g^{-1}$) such that $h_\ast(\omega) = dx \wedge dy$.

My question is: can we search for a solution $X_t$ that is invariant for $G$? In other word, if $\frac{d}{dt}\psi_t = X_t \circ \psi_t$, then $\psi_t(G) \subseteq G$?

I wish to know if there is a sort of "constrained Moser Trick". Suppose we have a planar grap $G \subseteq \mathbb{R}^2$, with $(0,0)$ as a vertex. Suppose to have some volume form $\omega = g_\ast(dx \wedge dy)$, where $g$ is a diffeomorphism of the plane that send $0$ into $0$.

By construction, the Moser equation

$\frac{d}{dt}\omega_t + \mathcal{L}_{X_t}\omega_t = 0$ with $\omega_t = \omega(1-t) + tdx\wedge dy$

can be solved, and we can find some $h$ (in this case $= g^{-1}$) such that $h_\ast(\omega) = dx \wedge dy$.

My question is: can we search for a solution $X_t$ that is invariant for $G$? In other word, if $\frac{d}{dt}\psi_t = X_t \circ \psi_t$, then $\psi_t(G) \subseteq G$?