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As predicted, changing XOOPS_URL to brilliantenergy.com allows it to work without the www. However, one get's the problem in reverse.
I think I will go without the www for XOOPS_URL in mainfile.php and then see if I can create an alias for www in my DNS. (Actually, I just checked the DNS and it appears there is already an A record for *.domain.com, so I guess that's not going to be enough. For the time being, I guess I'll add www back to my XOOPS_URL .)
I'm happy to hear other suggestions.