Ah -- the Ukrainian pains of asylum.
A couple of pointers about applying for asylum when you are already in the US. First, affirmative asylum (that's the kind you apply for when you aren't in the hot seat, i.e. in front of the immigration court!) must be done in the first year you are in the country. Sadly, the government does not want nor approve opportunistic "squatting" in the US -- hanging out here in the States until something bad happens at home, and then...poof! Asylum time! So, as for affirmative asylum, you have a great shot if you're in the first year of your presence a la United States. Otherwise...nada. See 8 U.S. Code § 1158 for information on asylum and the process of applying.
What may come open, and is designed for ye olde squatters as mentioned before, is relief called "TPS"-- temporary protected status. From time to time, the President deigns to designate certain places of the world as "particularly nasty and worthy of political consideration" -- i.e., those from that country may apply to stay in the US, even without valid status, for a period of time (usually renewable). Traditionally, this is reserved for South and Central American countries whose political base in the US is strong (wonder why?), but it is occasionally extended to others (say, Syria, where you know -- there is a war happening). The trick with TPS is that you MUST apply when the application period is open. While there are provisions for late filing, it is a tough road to hoe convincing USCIS that you meant to file even though you didn't. As Ukraine is not at war, and there is almost NO political Ukrainian vote in the US...this is not looking good :( See 8 U.S. Code § 1254a for the process of applying for, and information related to, temporary protected status.
Finally, for the wealthy...err...for those that feel up to a game of DHS/ICE/USCIS roulette, there is the concept of humanitarian parole. Usually this happens when someone is dying in your family (who happens to reside in the states) and the only option is to stay in the here. Occasionally, you can argue that is it YOU who is dying and should be allowed to stay in the states. That's an interesting argument, and by the time the damn thing is reviewed, you'll probably be dead. Which makes acceptance of the parole a little difficult. The key to humanitarian parole is that you need a really good brief, and it costs a lot of money to apply. The application to USCIS is free (send it to the local field director), the problem is that its a bear to write (think of this as an asylum on steriods). Don't let folks tell you it is the same as asylum. You can request this anytime when return to the country is nigh impossible. This makes extreme hardship of the 601A look like a regular picnic. You get the idea. See http://www.uscis.gov/sites/default/files/USCIS/Resources/Resources%20for%20Congress/Humanitarian%20Parole%20Program.pdf for information on the ICE/DHS/USCIS program on humanitarian parole.
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