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How can at least one of your names not be on your child's birth certificate?
To get an official birth certificate you go through the vital records office for the city or county in which the person was born.
Your question is not entirely clear.
If you are trying to get an official copy of your child's birth certificate in person or via the mail and the Office of Vital Records is demanding proof of identity then you have to provide it. The name of at least the mother HAS to be on the birth certificate, unless the mother gave a false name at the time. You just need to be able to prove you are the mother by providing ID. If you can't do that you have a big problem. If yes, you need to talk to an attorney about options.
If you have a photocopy of a birth certificate and need to get an official copy but the photocopy is of a document that doesn't show the names of the parents then the one you are looking at (without your names on it) is probably only an abbreviated birth certificate.
Many states started issuing the abbreviated birth certificates in the 1990s, with only the name of the person in question plus date & location on them. The full birth certificates - or at least a version with the parents' names - ARE available and you just have to ask for one.
I got copies of mine several times, just applied and paid, no ID was ever required, in Pennsylvania.
If your names aren't on the certificate then you can't get it.