= Solution
For the infinitesimal diffeomorphism generated by $X^\nu$,
$$
\delta g_{\mu\nu}=2\nabla_{(\mu}X_{\nu)},
\qquad
\delta A_\mu=X^\nu F_{\nu\mu}+\nabla_\mu(X^\nu A_\nu).
$$
Substitute these into the given first variation and integrate by parts. Symmetry of $E^{\mu\nu}$ gives
$$
\delta S=-2\int_M
\left(
\nabla_\mu E^\mu{}_\nu
+\frac12E^\mu F_{\mu\nu}
+\frac12A_\nu\nabla_\mu E^\mu
\right)X^\nu\,\operatorname{vol}_g.
$$
Boundary terms vanish by assumption. Since $X^\nu$ is arbitrary and diffeomorphism invariance says $\delta S=0$, the coefficient must vanish:
$$
\nabla_\mu E^\mu{}_\nu
+\frac12E^\mu F_{\mu\nu}
+\frac12A_\nu\nabla_\mu E^\mu=0.
$$
When the Maxwell equations $E^\mu=0$ hold, their divergence also vanishes and the identity reduces to $\nabla_\mu E^\mu{}_\nu=0$, the off-shell origin of stress-energy conservation.
Solved by gpt-5.6-sol high.
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