A couple of months ago I put The Egg next to a strong workshop halogen lamp in a garage and when I smelled melted plastic it was already too late and I got a nice soft-boiled one, too hot to even touch it with bare hand
After I managed to cool it down it turned out that the remote works just fine, while the key doesn't start the engine, with 'immobilizer failure' message on display. I called a local stealer (good one!) and after having been told the price for a new Egg, I burst into laughter and kept laughing until the guy hung up.
Apparently dealing with transponder problem (the remote works fine, remember) and having nothing to lose, I managed to open The Egg apart (man, I hope it was the last time in my life when I had to do that... and NO, it is not just about opening the lid as if you were about to change the battery, which comes off in 2 seconds, but disconecting the key from the remote). I then removed the transponder (a tiny black cube located it the key itself under a plastic cap). Virtually it looked intact, but must have been either frazzled or just deprogrammed under high temperature.
Next I took the spare key (the one without remote, but with working transponder) and removed the transponder from it (much easier this time, just had to lavarage and pull out a little 'drawer' in front of the key which holds the transponder in place). Not surprisingly, transponders in both keys turned out to be identical
When I have some spare time to go to a stealer I will ask them to try to reprogram the one that doesn't work (according to them, this option is 5 times cheaper than ordering a brand new key, but requires the transponder to be in working condition). Will let you know once I have it sorted out.