Solution
= Solution
Apply the product rule for the <Lie derivative> to $\tilde g=e^{2\phi}g$:
$$
\mathcal L_\xi\tilde g=e^{2\phi}\left[2\xi(\phi)g+\mathcal L_\xi g\right]
=e^{2\phi}(2\xi^c\phi_{,c}+\psi)g.
$$
The prescribed transport equation makes this zero, so \b[$\xi$ is a <Killing vector field> for $\tilde g$]. This is a conformal rescaling that absorbs the original infinitesimal dilation.