Cedric, yes, you may do so, provided you get married to your girlfriend. Contact experienced immigration attorney to assist you with the process.
Contact attorney Gintare Grigaite, Esq. at 201-471-7989, located in New York and New Jersey. Answers on AVVO do not constitute legal advice and do not form attorney-client relationship. Always consult an attorney for a legal advice.
There are a couple of different ways you can go about documenting your son's US citizenship, assuming you meet the requirements for transmitting citizenship. You can either apply for a US passport for him, or obtain a registration of birth with the US Embassy in Belgium. More information is available here:
For your girlfriend, as my colleagues have stated, you would have to be married before you can petition her for permanent residence. Also, as the name suggests, permanent residence is something she should consider if she actually intends on moving with you to the United States., rather than just visiting occassionally.
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REgarding your girlfriend, you cannot file any applications for her if you are not married and do not intend to get married. Regarding the child, there must be an acknolwedgment of paternity in writing and under oath, or paternity must be established by a court. You can apply for a passport for the child at a U.S. embassy and they will let you know what you need to submit.
The above is intended only as general information, and does not constitute legal advice. You must speak with an attorney to discuss your individual case.