Thank you
@esafeddie and
@P1LGRIM. Yes, EmuVariableUefi-64.efi is installed (not RC Scripts).
I have two AppleID's, a "primary" ID - created when Apple introduced .Mac services (1999?) and heavily used. Back in 2012, I created a "backup" AppleID - this account has been mostly dormant since. When I decided to try to get iMessage working on my (first) Hackintosh, I was a bit hesitant to use my "primary" account for testing this. So I reactivated my "backup" account, registered a credit card and tested from my MBP, iPhone and iPad's. Account is fine. But not for iMessage registration from my Hackintosh. Yesterday evening (CET), just before going to bed, I tried (for the first time) to register Hackintosh with the iMessage services of my "primary" account. What do you think .... SUCCESS. Again, many thanks to all.