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Do I actually have rights where my children are concerned?
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In US law you absolutely have rights regarding your children. What country in Europe are you in? I don't know European family law but I would doubt you have no rights there.
The main question you need to be concerned about is jursdiction, where will the divorce and custody actions take place. Divorce could take place in either, it just depends on who files first probably, but custody is almost certianly limited to the children's home state.
You're going to need at least a lawyer in Arkansas, and maybe a lawyer in Europe where you live if you want to file divorce there.
The Arkansas lawyer doesn't need to know European law, he/she's going to be fighting here for custody so you will be subject to the laws of Arkansas. Fortunately most states follow uniform international child custody laws (but I'm not sure with Arkansas, check with a lawyer).
As the biological parent you will certainly be entitled to visitation, and maybe partial custody (during the summer or other vacations). But you're going to need a lawyer to argue for you, and be prepared to spend a lot on airplane fees for you and your kids.
Depending on income you may have to pay child support, and maybe even some form of alimony.