There were bands (so called «VIA/ВИА») offically allowed to exist by authority of USSR. They played songs written by limited set of authors and was under strong control. Additionally, russian traditions prefers lyrics&vocals rather than music&beat. But despite all of this, they went funky sometimes.
Poyushie serdca — Davay pomirimsya.mp3
. The beginning is worth to listen to in context of groove.
Veselye rebyata — Chto takogo.mp3
. The most groove (even funk) starts since 2:20 although the beginning is rather good too.