Thank you for the help with the troubleshooting.
I followed JWWebster's advice and read the manual.
I found that, for the spinning problem, "if the clutch spins when the timer is set for spinning, the gearbox works OK" - a very important piece of information. The clutch was spinning in that setup in my washing machine. So, I moved my attention away from the gearbox.
Next, I found in this or another forum that "if the clutch gets hot when running in a spinning cycle, even for a short time, that means that the clutch is "sliding" and that is the reason that he washer does not spin". I checked the clutch in my washer after the spinning cycle and it was really hot.
I ordered the clutch.
I installed it today and the washer spins like a new one.
Thank you very much for all the help and assistance.
I still have one more problem, present just in one program "Normal Program", when the High Spin starts at the same time the washer also takes water. I have to turn OFF the timer and turn it ON again and the High Spin works as it supposed to. I thought I saw that problem reported by somebody but don't remember the solution.
I will appreciate an advice on this one.
Thank you again.