Iam a mother of two an I don't longer live with the father of my children and I was never married to him, but his name is on the brith certificate, dose he still have custody for them? I been separated for a year and 3 months and I want to know if I can move to an other city with out me getting myself intrable for talking them?

There is no custody order in place so whomever physically has the children has custody. You will not be violating the law to move away as long as you don't hide the children from him, but if he files a custody suit, you can be forced to return the children. You will want to have a custody order and visitation established BEFORE moving away.