Resetting PRAM did not change anything. Resetting SMC was the solution (and a very simple - just pull the plug and have a beer). Thanks for helping to get rid of the nightmare.
The question, however, is why does this happen so oftenly? It took me about 6 hours to check all potential culprits. The incidence that caused the problem was a simple breakdown of an application that has nothing to do with bluetooth. After that, my system has generated hundreds of false messages and indications that everything would function perfectly. My keyboard was accepted, some other devices, too. But I have not been able to change anything without attaching non-bluetooth devices. Before that I needed to use another computer to learn about it. Who, in **** owns, a battery of input devices and additional computers to hunt for the culprit?
Dear Mr. Apple: Think! But efficiently.