= Solution
Use flat target <spacetime> with metric $\eta_{\mu\nu}=\operatorname{diag}(-1,1,\ldots,1)$, and let $h_{ab}$ be an independent Lorentzian <worldsheet metric>. The <Polyakov action> is
$$
S_P=-\frac T2\int_\Sigma d^2\sigma\,\sqrt{-h}\,h^{ab}\partial_aX^\mu\partial_bX^\nu\eta_{\mu\nu},\qquad T=\frac1{2\pi\alpha'}.
$$
Here $X$ is the <string embedding map> and $T$ the <string tension>. Variation of $X$, followed by integration by parts, gives
$$
\delta_XS_P=T\int_\Sigma d^2\sigma\,\partial_a(\sqrt{-h}\,h^{ab}\partial_bX_\mu)\delta X^\mu+\text{boundary term}.
$$
Thus the embedding equation is
$$
\boxed{\frac1{\sqrt{-h}}\partial_a(\sqrt{-h}\,h^{ab}\partial_bX^\mu)=0.}
$$
The boundary term vanishes by periodicity for a <closed string>. At an open-string boundary it requires, for example, vanishing normal derivative for a <Neumann boundary condition>, or $\delta X=0$ for a <Dirichlet boundary condition>. Mixed choices give corresponding endpoint conditions.
Define the <induced worldsheet metric> $\gamma_{ab}=\partial_aX\cdot\partial_bX$. Using $\delta\sqrt{-h}=-\tfrac12\sqrt{-h}\,h_{ab}\delta h^{ab}$, metric variation gives
$$
\delta_hS_P=-\frac T2\int\sqrt{-h}\left(\gamma_{ab}-\frac12h_{ab}h^{cd}\gamma_{cd}\right)\delta h^{ab}d^2\sigma.
$$
The second set of equations is therefore the vanishing <worldsheet stress tensor>:
$$
\boxed{\gamma_{ab}-\frac12h_{ab}h^{cd}\gamma_{cd}=0.}
$$
For a nondegenerate induced metric this says $h_{ab}=e^{2\omega}\gamma_{ab}$. Substituting into the action gives $S_P=-T\int\sqrt{-\det\gamma}\,d^2\sigma$, proving the <classical equivalence of Polyakov and Nambu–Goto actions>. The auxiliary metric has not added physical metric degrees of freedom.
There are three important symmetry statements. <Worldsheet diffeomorphisms> change the parameterization, with $X$ transforming as a scalar and $h$ as a tensor; the integral is invariant. <Weyl transformations> $h_{ab}\mapsto e^{2\omega(\sigma)}h_{ab}$ leave $\sqrt{-h}\,h^{ab}$ invariant specifically in two dimensions. These are gauge redundancies. Locally, the two diffeomorphism functions and one Weyl function fix the three independent components of $h$ to <conformal gauge>, $h_{ab}=e^{2\omega}\eta_{ab}$. Global <worldsheet moduli> may remain and cannot be gauged away. Separately the flat-target <Poincaré group> acts globally on $X$ and supplies conserved target momentum and angular momentum, rather than another worldsheet gauge constraint.
In conformal gauge the embedding equations and metric constraints become
$$
(\partial_\tau^2-\partial_\sigma^2)X^\mu=0,\qquad
(\partial_\tau X\pm\partial_\sigma X)^2=0,
$$
or equivalently $\dot X^2+X'^2=0$ and $\dot X\cdot X'=0$. The <Virasoro constraints> must still be imposed: fixing the metric in the action before varying it would miss them. Residual conformal changes of the two null worldsheet coordinates are generated by these constraints. Classically these statements hold in any target dimension; cancellation of the quantum <worldsheet Weyl anomaly> is a further condition, not part of the classical variation.
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