I have an old PC - Intel Core 2 Duo 1,86 MHz, 4 Gb RAM, Video Nvidia GeForce 9800 GT 512 Mb 256 bit, and the game won't work :( even in case of sowtware rendering. When I start it, only a black screen appears and music, but the image does not appear at all. Help, please.

I suppose you could try running the game using eduke32. I'm not sure if it will help, but it helped me deal with some lag I had in my game. https://dukeworld.com/eduke32/synthesis/latest/?s=d&o=d&dir=eduke32/synthesis/latest

I used eduke32_win64_20190820-8043. I only needed the eduke32 exe. I dropped it into the Ion Fury folder and ran it. It detected the game just fine. Of course eduke wont use any older saves you may have so you'll have to start a new game. Any older saves will still be there when you run the game normally. I think it would be nice if someone could reverse engineer the game to run as a pure DOS game. I bet everyone could run that.

Ion Fury works with eDuke32? Isn't it running on this engine anyways or did the devs start from scratch with their own fork of the engine?

A DOS version of Ion Fury? I doubt that the limitations would work on a real DOS computer. I mean, it was/is even hard to run John Romeros DOOM Episode "SIGIL" on a real DOS machine. Ion Fury has so much detail and geometry, I doubt that such limitations would work on a DOS computer.

I had and also have some issues with Ion Fury's performance. But currently nothing gamebreaking. With Vsync enabled, I have sometimes 30 fps, which is okay for me. When the game was in Early Access, the performance was much worse for me. Level 3 of the preview campaign was hardly playable for me because the fps went crazy. Changing the options did not help. But that was way back in 2018, the engine is now much better working for me. Props to the devs!

I have a Core i5-2500k and a GTX 970. Without Vsync, my fps are great.

Ion Fury uses a modified build of EDuke32. It's made largely by the same people, too.

And they've stated several times that it's not a good idea to use the normal EDuke32 binaries that have not been compiled with the FURY=1 option to run the game.

Makes sense if I recall how the performance was for me in 2018 for this game. I remember Digital Foundry talking about the work on the engine, that a lot will be changed and optimized. Correct me if I remember wrong.

The one I linked obviously has Fury=1.
But I think we got off topic. The OP wanted help with black screen.

https://www.gog.com/forum/ion_fury/ion_fury_troubleshooting_thread

This person mentions a black screen issue too and a possible solution.