Lithuanian Grandmaster. Baltic Champion in 1985. Lithuanian Champion in 1983, 1986, 2001, 2008 and 2012 and the Estonian Champion in 2001 and 2002.
Best results: Winner or shared first in Ostrava 1992, Prague 1993, Marbach (GER) 1994, Bonn GSK 1995, Parnu 1996, Capelle la Grande 2002, Staufer Open, Bristonas Open, Charleroi IM and Lausanne Open 2003, Bad Worishofen Open, GM Bogny 2004, Norwegian Open Championship and Gausdal IM 2005, Jozef Kocan Memorial (POL) 2006, Panevezys (LTU) Open 2007, Moss GM 2009. He also came second in Bonn GSK 1993, Mezezers Zonal 2000, Bobigny (FRA) 2003, North Sea Cup 2006, Svein Johannessen Memorial 2008 and in Gausdal Open 2008, Bad Worishofen Senior 2016.
Kveinys was a participant in the 2001 World Championship and in eight Olympiads between 1992 and 2008.
Lithuanian Grandmaster. Baltic Champion in 1985. Lithuanian Champion in 1983, 1986, 2001, 2008 and 2012 and the Estonian Champion in 2001 and 2002.