= Solution
The <Levi-Civita connection> has symmetric <Christoffel symbols>, so antisymmetrization removes its connection terms. Hence the <Papapetrou electromagnetic field> is
$$
F_{ab}=\nabla_aV_b-\nabla_bV_a=2\nabla_aV_b,
$$
where the last equality uses the <Killing equation>. As a <differential two-form>, $F=dV^\flat$, so the identity $d^2=0$ for the <exterior derivative> proves $\nabla_{[a}F_{bc]}=0$. Contracting the preceding <second covariant derivative of a Killing vector> gives
$$
\nabla^a\nabla_aV_b=-R_{bd}V^d,
\qquad \nabla^aF_{ab}=-2R_{bd}V^d.
$$
A <vacuum spacetime> here has $R_{ab}=0$, with no <cosmological constant>. Therefore \b[both source-free Maxwell equations hold]:
$$
\boxed{\nabla_{[a}F_{bc]}=0,\qquad \nabla^aF_{ab}=0.}
$$
This is a test <electromagnetic field> on the given <vacuum spacetime>; its own <stress-energy tensor> is not included in that background's <Einstein field equations>.
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