The Russian song is called Sing, Swallow, Sing. (Swallow as in bird not in gulp)

It is not Ochi Chyornia: it is the same tune as the music hall song "Two Lovely Black Eyes."

I do not know where the tune of "Two Lovely Black Eyes" originates. All I know is that the Russians say it is one of theirs. If you trawl through Europe you might find that melody all over the place. It might be Italian; It might be Russian; it might come from somewhere else.