I just found a bargain (New York to Kyiv) from Aerosvit.
Oddly, when I checked aerosvit.com, I didn't find this fare on their list of promotions.
Depending on the day of the week, you can book roundtrip from JFK to KBP from the first half of March, with returns from anytime in March until at least early April (I was doing a bunch of searches on kayak.com, so I was trying to understand the nature of the elephant by blindly feeling it), for $398 including fees and taxes.
These are direct flights, and about as comfortable as most economy transatlantic services.
If you want to look for these fares, I recommend using kayak.com, and selecting the flexible dates +/- 3 days option.