LSGA (Laboratory for Sports and Racing Cars) is probably only partly fitting, as it was a VAZ child. Can't say it was a VAZ sports department, as there is only one car that is mentioned in relation to LSGA I could find - Lada 2105 LSGA, while before and after there were other cars by VAZ related enterprises (VAZ structure is really a complete mess for me). It was a 2105 with 2-door body, McPhersons in both front and rear, 16-valve 1730cc engine (highly modified 2106 unit) capable of 150hp, built around 1984. Info is scarce, photos - even more. Have not survived.
Tavria ERF-Mobile was a rally prototype built around tube frame and VAZ 2106 engine with ZAZ Tavria-ish body. It was designed in current Estonia in 1986 at sports & technical club of ERF (Estreklamfilm - at work those guys produced TV ads and so on...) by the club's director - Kalle Keel of TARK (Estonia formulae) fame. It was a common practice in USSR that factories and various collectives had their sports sections (in everyday life many Soviet racing drivers drove taxis, ambulances, even army trucks), so no wonder that movie making guys have eventually come up with a car
2106 engine of 1568cc, 140hp, behind the driver. Independent suspension, weight around 800kg.
I've also seen a picture of ERF rally team with three 2101 Ladas (those were hardly of their own preparation).
ERF-Mobile was said to be seen at some auto show in Tartu in 2002 and is probaby in private collection now (no proof).