Here's an official early concept image:
The institute will develop the preliminary design in partnership with several Russian companies, including 'Sukhoi' and 'NPO Saturn'.
One of the primary issues with Supersonic Transport Planes 'SSTs' (apart from cost) has been their sonic boom (or more specifically the sonic booms they generate while in supersonic flight):
which has resulted in SSTs being prohibited on going supersonic over populated areas. Additionally there is also the higher engine noise levels on SSTs and the significantly higher thrust they need compared to other aircraft. SSTs generally provide poor performance in slow flight - they have poor lift/drag ratios at subsonic speeds (unless swing wing technology is employed) and burn more fuel in these circumstances. Supersonic aircraft are designed to go fast, if they are not going supersonic then they are pretty far from being cost effective.
The solution Concorde applied to these problems was to wait until the aircraft is at high altitude and over water before reaching supersonic speeds. However in the end this is just a technique to go around the problem and doesn't address it - supersonic flight over populated areas was still not possible, thus limiting the use of the Concorde and meaning the aircraft can only be used cost effectively for routes over oceans.
If TsAGI can develop a quiet supersonic business jet it would be a great achievement, which would in turn encourage more work on developing effective SST airliners.
In Russia, since the 90s, both Tupolev and Sukhoi have been involved in creating designs for supersonic business jets.
Research so far has indicated that it is certainly possible to design a quiet supersonic jet - there are even theoretical designs that do not appear to create sonic booms at all! Careful shaping of the fuselage of the aircraft can reduce the intensity of the sonic boom's shock waves that reach the ground. During 2003, a Shaped Sonic Boom Demonstration aircraft was flown by NASA, which demonstrated the capability to reduce the sonic boom by about half. There has also been 'Quiet Spike' and even designs involving causing the shock waves to interfere with each other. Just lengthening the aircraft (without significantly increasing its weight) seems to reduce the boom intensity. With the increasing power of computer-aided design it has become possible and much easier to develop and test such designs.
The only SSTs to see regular service to date have been Concorde and the Tupolev Tu-144. It would be great to finally start seeing new successful types being developed and created.