= Flat conformal Killing integrability
{title2=$\partial_\beta\partial_\gamma\omega_\alpha=(\delta_{\alpha\beta}\partial_\gamma\rho+\delta_{\alpha\gamma}\partial_\beta\rho-\delta_{\beta\gamma}\partial_\alpha\rho)/d$}
For a flat <Conformal Killing vector field>, let $\rho=\partial_\mu\omega^\mu$. Differentiate the <Conformal Killing equation> in three index orders, add two equations and subtract the third to obtain the displayed identity. Tracing gives $\Delta\omega_\alpha=(2-d)\partial_\alpha\rho/d$, whose divergence implies $\Delta\rho=0$ for $d>1$. Taking the divergence of the original identity gives $(d-2)\partial_\beta\partial_\gamma\rho=-\delta_{\beta\gamma}\Delta\rho$. Thus $\rho$ is affine and $\omega$ is at most quadratic when $d>2$.
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