Can I obtain non parental visitation?

A recent paternity test revealed that I am not the biological father of my son. I have never been married to the mother, who lives with another man (not the biological father either) and has 3 sons whose fathers are unknown. From the moment she told me she was pregnant with my son I helped her as much as I could and I was there when she delivered (a little over 3 years ago). Although on the birth certificate she listed father unknown (without my knowledge), I helped her financially every month. My son would come over every other weekend and on holidays. Since all the child support I provided was undocumented, we decided to do this through the proper channels. The paternity test required for this (my son is over 3 years old) revealed I am not the biological father. Now the mother wouldn't let me have any contact with him saying it's not fair for her other 2 sons who have no father. I love my son and still want to be part of his life. I am a single man, early 30's and have low income.