Hi @Ivan Renoud-Grappin ,
Yes, this is true, once you have installed a Gateway, its Management Group affiliation cannot be chnaged. You need to reinstall it.
When you install it again, consider that on the Gateway you have a certificate, so that it can talk to the "old" management group. In case you re-install it, your GW needs to talk to the "new" management group and this might require installing and importing (MOMCertImport) a new certificate.
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Regards,
Stoyan