After about one year, I decided to put online some photos which show how I repaired my old travelmate laptop, hoping they will be useful for someone who had the same problem (or who just wants to see how a naked laptop is ;)).
The problem I had was a power connector which, after some years of usage, started to slightly move from its position until it broke the welding with the motherboard. As the contact with the mobo became dangling, it started to charge/discharge my battery continuously (in fact, it lives for about 5 minutes now) and then to let the computer switch off when batteries were too low.
Fortunately, no component was really broken so it was quite easy to repair: the hardest part was to realize how to open the notebook and reach the part I needed to solder. Here the whole process is described through photos.