What was the 500 BSA I had in the 1950's?

When I was a teenage (and as a small kid) I hung around an AMA motorcycle racing shop in the 1950's. and the guy who raced did so for a BSA dealer. When I got my license he said he had a BSA 500 Star Twin that needed rebuilding and if I gave him $50,00 and bought the parts he would help me rebuild it. It was titled as a 1953, but had a swing arm frame, also it was not a street bike but a factory racer so we had to wire in a head light and tail light for the street. I rode it several years and it would run with and out run many new 650cc bikes, plus out handle them. WHAT was this I had? (My old friend in Texas still has it).