Transformation that keeps Poisson process invariant in Lie groups

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As we know that the Brownian motion is invariant under orthogonal transformation ,

*see article of Bernt Oksendal for more details : **https://doi.org/10.1007/BF01045159**.*page 215 corollary 1.

My question is : what is the transformation that keeps Poisson process invariant ?

Means that,do we have a theorem or article that answer this like for the case of the Brownian Motion?

Thanks :)