I sent a email to you earlier today re you post in Ukraine forum.
Surname Zawertany was originally ??????????. Closer transliteration/sounds-like would be Zavertaniy.It means PERSON WHO RETURNS.It is very likely a Cossack surname.
Records show that the surname change was ZAWERTAN.
As per Ancestry records, the Zawertany were from village Buzovka in the Kyivskaya gubernia.TODAY the selo/village Buzovka/Buzivka still exists w/appx 2,000 residents. It is located in the Zhashkivskij raion/district of the Cherkasska oblast/region.