Hi. So, today I connected the burner to the pentium 1 mentioned above, and..... it surprisingly worked! Some PC, with 32 MB of memory and CPU (133Mhz) that don't even has a fan, runs it perfectly, while my much upgraded for it's age P3 doesn't!
Than I took it's 250 watt, old, power supply unit out, and connected it as a separate power source for the burner on my PC. And finally it WORKED!!!
Nitestick, you really deserve respect on this one. While you mentioned PSU for the first time, I tried disconnecting 2 hard discs, one CD burner and 2 big fans, what theoreticaly should release much more power for the burner, and it didn't work, I was prety sure the PSU isn't the problem. But you insisted on it, and was right.
It seems like DVD's laser is very power consuming thing, so the drive tryed to activate it and kept failing all the time what appeared from external point of view as it "lost a track".
Anyway, now I need a new power supply. But since I plan to buy a new PC (finally!) in about 6 months, I'll probably will try to save the money and find some old PSU and connect it the way I did today. It will be a bit messy to turn on another PSU for the drive everytime PC starts, but maby it is a way to start 2 PSUs with a single button, but I'll keep it for another topic once I find a PSU.
And thanks to everyone for assistance!!! As long as Tech-Forums exists there is no need for PC technicians.
-AndreiD