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Vector fields $X$ and $Y$ commute on a closed set $K$. Do there exist commuting $\tilde X,\tilde Y$ with $\tilde X=X$ and $\tilde Y=Y$ on $K$?

No, already not possible for $K = \{0\}$ a single point. Take e.g. $X = \frac{\partial}{\partial x} + (x^2 + y^2 + z^2)^2\frac{\partial}{\partial z}$ and $Y = \frac{\partial}{\partial y}$. These ...

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