Battery Problem

[RichInFla]

This has happened twice so far, and I've searched through the posts but can't find anything relating to what's happening. Might be a little long, sorry. First, no new equipment installed since I bought it. So, about a month ago, the bike sat for two weeks and wouldn't start, so I bought a charger, charged it up, and the next morning it started right up. Rode for about 3 hours. This weekend marked the end of another two week "rest" for the GZ, so Friday night I put the charger on it. Started right up Saturday morning and I rode for about 3 hours. This morning, went to go for a ride, it started right up, but I was coming to a light after about an hour of riding and it just shut off. Electric start did nothing, tried push starting (which I was successful with about a month ago), but it wasn't happening this time. Luckily, my HD riding neighbor had a truck and ramp, so he came and got me. Put the battery on the charger when I got home and it was at 16%/2A. :??:

He said it might be my "stator" (he pointed to a little metal thing with wires coming out mounted the frame right under the seat) or "alternator" (don't know where that is or if the GZ has one !!). He said the battery is just supposed to start the bike, but mine was "running" off the battery, draining as I was riding. He has a riding buddy who has the same problem (different bike). He said he can ride for about 4 hours, then it'll shut off and just won't start.

Okay, GZ Wizards, I'm in need of your infinite wisdom, I can't afford another trip to the "shop" after just purchasing tires and brake pads ! Is there anything I can try myself, I can follow instructions !! Thanks all !!