I cannot setup my @mac.com or @gmail.com account in my mail

For some weird reason I cannot setup my @mac.com account in my mail, all the other accounts are working fine. I suspect it has something to do with the synch services but I cannot find a fix.

All this happened in a migration from and old Mac Pro to a new Mac Pro with retina display using the time machine backup, once done my @mac.com account was present in the Internet Accounts within Systems Preferences but not in the list of accounts on my Mail.

Its doing strange things, all mi company email accounts which doe not need to synchronize are present in my Mail accounts and where added through this app and are working fine, if I try to add the @mac.com or @gmail.com accounts through the Mail application they will not appear in it as created. If I add these account in Internet Accounts withing Systems Preferences they will not appear in the Mail application (even if I turn them on as email).

Any ideas?

Thanks

Alonso