OK, it's a little weird. My wife, thinking that someone had damaged the cable of her Macbook, had unplugged it. She then ran the Macbook until it shut down by itself. When I got home, I tested the cable and everything went fine, except that the light did not light up when I plugged it in (the first time , the green light is on briefly, but never since.).
I do not know how to solve this problem. The Macbook does not turn on, so it is impossible to get information about the system or reset SMC. I know it's an A1278, probably from 2011/2. I have cleaned the contacts of the cable and Macbook. Also, disconnect and reconnect the battery. Until now, no joy. All ideas / suggestions would be welcome.