Burning pain in lower back that comes goes for 3 weeks. had a uti infection gone now. pain in lower back still there. comes at random. ?

Could be infection: still, or musculoskeletal pain. If there is more odor and frequent discharge unrelated to frequency of menstrual flow, then it's likely to still be an infection. If you use your back a lot or strain it easily, then have it checked out by a healthcare provider to treat properly so it doesn't keep coming back. Other possibilities include kidney stones or pregnancy -- check a pregnancy test.