This problem is not known to affect the ET or LX models. The GT60 and GTV models, since they draw directly from the GTS product, also share the habit of wobbling.
Generally, the GT and GTS steering will wobble slightly (some would call it a head shake) when decelerating through 30 mph (approximately) with one hand removed from the handgrips. Not all of them exhibit this behavior, although it's likely that all of them are capable of exhibiting this behavior given a few changes in parameters.
Some of the things that are known or thought to affect the presence and severity of wheel wobble on the GT and GTS are: tread pattern, rider weight, cargo weight, shock preload, tire inflation, insufficient bar-end weights, and wheel balance of the front and rear wheel. Any or all of these factors might combine in some strange way to cause your scooter to wobble slightly when one hand is removed from the grips.
Probably the single biggest factor determining the presence or absence of the wobble is the tread pattern of the tires. The GTS comes equipped from the factory with Sava tires, and when so equipped it doesn't typically exhibit the wobble. It is thought that the straight profile of the Sava tires minimizes or masks any tendency to wobble that the GTS might have. Typically, as soon as better tires are installed on the GTS, the wobble appears immediately.
Generally, this wobble is not considered dangerous, although it can be disconcerting for someone who has never experienced it before. If your scooter wobbles uncontrollably (i.e. with both hands on the handgrips) or wobbles at higher speeds and while accelerating, you probably have some other problem, and we strongly advise that you take the scooter to your dealer for service. Some possible factors that could cause severe wobble might be loose steering bearings, frame alignment, or fork alignment.
If you have the garden variety wobble, however, your dealer may be reluctant to attempt to address the issue. Don't blame your dealer, it's a difficult problem to solve, and involves changing many things in various combinations to try to find that sweet spot where the problem no longer happens. This is one of those things where you really have to ask yourself how much time and effort you're willing to invest trying to track down what amounts mostly to a slight annoyance.
If your scooter has the symptoms described in this article, here's a series of experiments you can try:
1) Seat Position. Try shifting your position forward and backward on the seat to determine where wobble is most pronounced.
2) Tire Inflation. Underinflated tires are more likely to wobble than overinflated ones. Make sure your tires are at the factory-specified levels and see if the wobble is still present. If so, try overinflating your tires a bit (not a lot) and check again. This goes for both the front and the back tires.
3) Top Case. If you have a top case, remove it as an experiment to see if the wobble goes away. Also make sure you have the heavier bar-end weights on your handlebars. These come with the Piaggio topcase, but not 3rd party top cases.
4) Wheel Balance. Have both your front and back wheels balanced properly. Marc Parnes Wheel Balance Video Demonstration
5) New Tires. The Sava tires that come stock on the GTS mask or eliminate the wobble, but they're not very well regarded in all other areas. In particular, they catch on rain grooves, which can be much more disconcerting than the steering wobble. The new tires on the GTS 300 are Pirelli GT23, front and GT24 rear DO actually seem to eliminate the wobble, according to rider reports. They are a semi-sporting tire with excellent grip.
6) Rider Weight. This is often the hardest thing to change, but it does seem that heavier riders experience the wobble more often than lighter riders.
If you do have some success at finding a solution, please post a note indicating what worked for you. Collectively, we might at least get closer to knowing what combination of things works to eliminate the problem.