10.6.8 Snow Leopard Mail

In the middle of June 2017 my mail on my iMac repeatedly asked for the password.
"Enter the password for the user [myname] on icloud.com IMAP server "imap.mail.me.com"

Note - I can access, with password my mail online through icloud with no problem.

The mail on my Samsung phone stopped working at the same time.

I have read every article and Q&A about this problem and whatever the solution suggested, I have tried and failed.

I have reinstalled the application from the system disc, encountering the "this version cant be used with this OS". I solved that problem.

I've created new accounts, phoney accounts, deleted old accounts, I've tried every combination of everything in mail configuration/prefs that's been suggested everywhere on the net.

Nothing works.

Someone, please help.