Is it possible to leave the mother off of a birth certificate?

Fathers are often left off because the mother doesn’t want him involved or want him to know about the child. Maybe she doesn’t know who he is or maybe the father wants nothing to do with the child. I’m sure there are other reasons as well. Point is, it can be done.

The identity of the mother is obviously known, but what if she doesn’t want to be on the birth certificate? She doesn’t want to be identified or associated with the child? For example, she was going to have an abortion, but had the child only because the father wanted it and agreed to release her from any obligation.

Should mothers have the same possibility of anonymity as fathers?