Guys, some of you know that I have a Nintendo VS Unisystem. I bought the game a few years ago with a container groupbuy. The cab is in very good cosmetical condition, very solid woodwork, but ........ it's

Yep,
green, acid green (gifgroen!). Painted by some Americano.

Still, it's a very nice cabinet and I have two great games for it. Dr. Mario and Vs. Super Mario Bros.
The cab has only one real damaged spot, fortunately at a place where new side art will hide it:

Hey,
orange! 
Of course, this cab used to be a DK Jr, according the serial plate at the back of the cab. Fortunately, I always thought the acid green paintjob was technically a poor job, the paint feels sticky and has a strong orange-skin effect (you know, the effect that women hate on their skin). Probably it would also have a bad bonding with the original orange layer.
So, I tried to peel away the green paint, just with my fingernail. And it peels away very easily. This spot was done in a few seconds:

I tried a few more spots on the frontpanel



Yep, everywhere the nice orange color appears. Seeing the excellent condition of the cab I now have high hopes that the original orange skin is everywhere.
But how to remove the green paint from the whole cab, without damaging the layer beneath it? I can't do it with my fingernails, so who has the golden tip for me? What miracle stuff do I need to use?