thats an ignition coil you have. Search ebay for "scooter generic ignition coil". Should be able to purchase for £10 from a UK trader. Likely they are available for cheap off ebay.fr too.
There is no repair as the coils of copper wire are "potted in" with the black plastic you see the high voltage lead coming out of.
It may be that the magneto has failed though, it is not so common for an ignition coil to fail but not unheard of either.
I see that it is the usual cord pull starter. Have you check for a spark? If not the usual way is to remove the spark plug and fit it to the plug cap. Place the metal body of the plug snugly against a metal part of the engine and pull the starter cord as though you was firing her up. Look for the spark in the electrode gap of the plug.
Don't try in bright sunlight you won't be able to see it, it is tiny. Should be decent blue end shade of purple, won't be very strong but should be clearly visible.
Maybe that was your basic test in which case I am telling granny how to suck eggs! If its the ignition coil then cheap and easy if its the magneto coil not so cheap. Those two terminals that the magneto connectors push onto, if you put test meter probes onto those with the coil itself connected then pull on the starter cord the meter should show a reading on the ac range. It is only a brief pulse for each rotation of the crank so no value is to be looked for, just change. Older analogue meters are better for such a test as the needle will deflect at the slightest change in voltage.
Let me know what you have tried and whats been confirmed and how it was.