= Solution
In a <stationary spacetime> chart, take $V=\partial_0$. Its squared length is $g_{00}<0$, so $V$ is a <timelike vector>. Since its components are constant, the <Lie derivative of a tensor field> obeys $(\mathcal L_Vg)_{\mu\nu}=\partial_0g_{\mu\nu}=0$. For the <Levi-Civita connection>,
$$
(\mathcal L_Vg)_{\mu\nu}=\nabla_\mu V_\nu+\nabla_\nu V_\mu,
$$
so $V$ satisfies the <Killing equation>.
Conversely, a timelike <Killing vector field> is nowhere zero. Apply the <flow-box theorem> to choose coordinates with $V=\partial_0$. The <Killing equation> then gives $\partial_0g_{\mu\nu}=0$, and $g_{00}=g(V,V)<0$. Thus \b[local stationarity is equivalent to the existence of a timelike Killing vector field].
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