To make matters worse, nearly every US-spec modern Vespa will suffer side effects from even a single overfill. On carbureted models in particular, overflowing gas will result in poor performance: difficult starts, rough idle, or intermittent stalls are all common symptoms of a recent gas overflow.
This problem applies to all Piaggio LEADER engines, which are used in the ET, GT, and LX Vespas, as well as several non-Vespa scooters.
The problem lies in the fuel filler neck, which has a hose to vent evaporated gases from the gas tank back through a closed system and into the carburetor. The purpose of this system is to reduce emissions by making sure that gasoline evaporation does not escape into the atmosphere, and is instead recycled into the engine.
If liquid gasoline reaches the evaporation hose inlet in the filler neck, either through overfilling or the expansion of the gasoline on a hot day, then liquid gasoline can enter the evaporative system and stay there. This prevents the carburetor from sucking air into that vacuum port, which in turn can cause your scooter to be hard to start. It can also cause it to idle roughly or stall altogether, depending on exactly how little air is making it through that vacuum line and into the carburetor.
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If you intend to leave the hose disconnected permanently, it would be prudent to affix a small foam filter to the hose inlet on the carburetor to prevent dust or debris from entering through this hole. Under no circumstances should you completely block the inlet on the carbeurator. Another possible solution would be to leave the evaporation hose connected to the carburetor but to fit a hose-T inline to the evaporation hose, again with a small foam filter on the end of the T to allow air to pass into the hose.
Gasoline can seep into the air filter foam, preventing the normal intake of air through the carburetor. The solution for this problem is to remove the air filter, wring out the gasoline, and wash, dry, and re-oil the air filter as per the user manual.
It's also possible that gasoline has soaked into a charcoal canister that lies between the carburetor and the filler neck, and this too can cause the aforementioned problems. If removing the evaporative hose from the carburetor doesn't fix the problem, and cleaning the air filter doesn't either, you may have to consult a Vespa or Piaggio service center for further assistance.