Which part of Russia does your family come from? The Biegun family I am related to originally comes from the Kavkaz, near the Sea of Azov. Actually the Biegun name is well-known in Russian history. The Bieguny (plural for Biegun, meaning "runner" or "fugitive") were so-called for their anti-State, anti-government feelings, and fled military inscription, etc. See RUSSIA OF THE RUSSIANS by Harold Whitmore Williams. My family's last name has had the "off" ending added in the U. S.
There may or may not be a link between our side of the Begun or Biegun family and yours, but it's fascinating unravelling the threads!
Tatiana Begunoff