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In this paper the authors consider an approximate Killing field $\chi$. It vanishes on a given 2 surface and its first order part is given.They say that if it obeys the Killing equation $\chi_{a;b}+\chi_{b;a} = 0 $ then its second order part vanishes. Do you understand why?

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Zhen Lin shows here that if a vector field $\chi$ obeys the Killing' equation $\chi^i_{;j} +\chi^j_{;i} = 0 $ then $\chi^a_{;bc} = R^a_{;bcd} \chi^d $everywhere.
As $\chi$ vanishes at p, we have at this point $\chi^a_{;bc} = 0. $ The second order part of the taylor serie being quadratic in the covariant derivatives, it equals zero. Raf Guedens who is a co-author of the initial paper gives many details in (https://arxiv.org/abs/1201.0542)

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