Thank you for so soon reply. I have neither domain, nor a server. I just configured exim4 to work with mutt and bugzilla. In Evolution there is an option to send mails with sendmail that sends mails with exim that is running on the system level. There is no such an option in Thunderbird but I guess there should be a work around.
This is my /etc/exim4/update-exim4.conf.conf:
# This is a Debian specific file