In Becker & Becker, Schwarz book String Theory and M-Theory, page $40$ is stated that after choose the conformal gauge $h_{ab} = \eta_{ab}$ in the Polyakov action with background field $G_{\mu \nu}(x)$
$$S_P[X,h] = -\frac{T}{2} \int \text{d}^2 \sigma \sqrt{-h}h^{ab}\partial_a X^\mu \partial_bX^\nu G_{\mu \nu}(X) \tag1$$ there is a residual symmetry in the action.
For infinitesimal diffeomorphisms $\sigma '^a (\sigma)= \sigma^a + \xi^a(\sigma)$ and Weyl transformations $h'^{ab}(\sigma) = \eta^{ab} +\Lambda(\sigma) \eta^{ab}$ that obeys
$$\partial^a\xi^b +\partial^b\xi^a = \Lambda \eta^{ab}, \tag2$$ i.e. conformal transformations that cancels with Weyl transformations, actually leave the worldsheet metric, and hence the conformal gauge choice invariant. This residual symmetry gives rise to other gauge choices. Two famous choices are the static gauge and the light-cone gauge.
My problem: My advisor asked me to search in literature and find that the static gauge choice $X^0 \propto \tau$ is not allowed for arbitrary backgrounds, i.e. curved ambient spacetime with metric $G_{\mu \nu} (X)$. After spending some time searching in web about this topic I didn't find anything helpful. Can someone recommend some resources or explain it to me?
EDIT I've found this article that treats static gauge for arbitrary static metrics $G_{\mu \nu}$. Presumably the author motivates the choice of this gauge considering that the energy
$$E \propto \int_{0}^{2\pi} \text{d}\sigma \partial_\tau X_0$$
is conserved, due to Noether's theorem. Anyways, this has not yet explained all the stuff to me.