Hi Ran Man,
1) What is the name you picked?
2) What industry is the company that is using your name in?
3) I have a computer game company (Flat Cat Studios), that is also the name of a small rock band. Since the Industry's are different that's cool, legally. My IP lawyer is part owner and did the company name registration.
4) My last company we named we used a Greek word "Askisis" which mean exercise. That has worked out well no hits in the USA trademark office on that one.
Take a look at
www.uspto.gov do a trademark search.
Ken