ok, maybe it was " kalinki" ?
in russian we have a song , EVERYBODY knows, it is " kalinki malinki" , but usually people sing it in bars after some glasses of vodka

)))
also in Bulgaria there is sth like that, " kalinki kato balerinki". i don't know whether it is a song or no, but there is sth like a small poem, maybe they sing it too, i just can tell that the contents is almost the same.
if u still search, maybe it's this. anyway i don't know anything more, "belinki" exist in russian, but "kalinki " we sing

)))) belinki is a sort of grass or sth, an archaism, very old word.
i'm glad if i can help u somehow, really

))) if u need my help, u can write, i'll try to do anything possible. Alia