Zero-data identities on the plateau

For $\xi ,\zeta \in {\mathfrak{g}}_U$, the fields $\xi$, $\zeta$, $X_{\xi }{\varphi }_0$, $X_{\zeta }{\varphi }_0$, and ${X_{\xi }}^2{\varphi }_0$ vanish on a neighbourhood of $\Sigma$. Accordingly ${\mathsf{E}}_0\square \xi =\xi$ and ${\mathsf{E}}_m(\square +m^2)(X_{\xi }{\varphi }_0)=X_{\xi }{\varphi }_0$. The relevant fact is the vanishing Cauchy data; compact support in $U$ is the sufficient condition declared on the sheet. For a parameter not vanishing near $\Sigma$, ${\mathsf{E}}_0\square \xi =\xi -H_0[\xi ]$, where $H_0[\xi ]$ is the homogeneous solution with the data of $\xi$; such terms would have to be included in the transformation law and are excluded here by the support condition, not by a universal identity.