Infant birth certificate has mother's name only

Mother wishes to travel to China with her son to visit her mother. Son is 13 months old. Mother is naturalized US citizen, son born in US. Father is natural born US citizen but does not have custody of child and was never married to mother. Child's birth certificate has only mother's name on it and son has mother's family name. Will this kind of birth certificate be acceptable to obtain a Visa for the son for a 30 day or less visit? Also, does the mother have to bring a letter of authorization signed by the father or will a court order allowing her to visit with her son be acceptable? Thanks.