This is a problem I see quite often in our South Florida Community. (Un)fortunately, South American immigration laws are actually quite protective of minors and they do not usually make exeptions for a single parent that cannot prove their permission to travel with their children. In order to secure permission to travel your choices are as follows:
1) find the father of your children; hire a PI; make phone calls; track him down to get his written approval to travel OR
2) file a paternity action; you will still need to locate him however in the mean time you may be able to ask the court for permission to travel OR
3) if you cannot afford a private attorney go to child support enforcement / department of revenue and start an action for child support; if he pays taxes they can find him. When they find him you can then ask him for permission.
There is no way to tell how long the court process would take as there are 101 variables that contribute. The second and third options are obviously the more complicated ones, and given the information you state, it is advisable to speak to a local attorney to explore this in detail.
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