To have lunch = having lunch, and should be accepted. Besides, the meaning, in this case, is perfectly identical. On this website, I was told the infinitive could be translated either way. Also, for the record.. in English, "by one's self" and "alone" mean the EXACT SAME thing.. I realize there are more ways to say "alone" in Russian, but that doesn't make the English word "alone" mean anything different than "by oneself."