Wish I could tell you how to wrench on your Peugeot Looxor. My expertise is on Harley-Davidson. However:
Check the compression. Small engines (two-strokes are the worst) are of course under undue stress, so it is common to suffer failed rings and low compression. Check your service manual to find the proper cylinder pressure (most often, about 125-150 PSIG).
It kick starts with an effort and will not start with the electric. Clearly, the system was properly wired at the factory. Know that the electric start will only push the crank around at 45-55 RPM on 100CC bikes, and the kick starter will push it around at 75-100 RPM. Again, compression is the most likely suspect, but check the fuel-air mixture, spark plug gap, and voltage to the spark plug with an oscilloscope (if you can find one with a 20 KV range).
The final guess in the dark is that your battery is not properly charged. A 100CC bike will draw about 25 amps from the tiny battery to start w/ electric, and it may not have enough juice left over to power the electronic ignition.
Hope that helps, but know that you will not find that misery on a Harley.
