It looks to me like the right axle bearing fried (horrifically overheated) and lost some of the needle rollers inside it. The interesting thing about those videos is that the rear wheel jumps out of alignment on the second revolution of the wheel, not each revolution. That indicates to me that what is left of the bearing is still rotating (probably just the needle retainer now and <maybe> a needle or two) and causing the oscillation to be out of time with the rear wheel. It also looks like the hub side of the rear wheel is supporting the rear wheel and causing it to turn pretty steadily and straight until the side- force from the brake application pulls the wheel away from that axis of rotation. I am not sure if this message will translate into Italian gracefully but I do not know how to make it much simpler and keep the meaning intact. ??
That bike is going to need a bunch of parts, perhaps including parts or even an entire new drive hub depending on how hot it got in there. The right axle mount is clearly shot as it badly overheated when the actual failure (I think bearing failure) happened. No doubt a new axle will be needed.
The usual mode of failure for wheel bearings is that the seal fails to seal, then contaminates enter the bearing itself and finally the bearing fails (rather than wears out). My 2008 C-14 is in need of a new rear wheel seal right now as a matter of fact. The wheel bearing is still fine but it won't stay that way for long unless sealed from water and road dust. By the way, I have replaced the front wheel bearing seals twice already on this bike and it is coming up on needing a set of front seals again, probably next summer.
I am not much for yelling about safety or becoming overly concerned about reasonably normal noises, wear, etc. but that is WAY, WAY outside of the normal envelope and needs to be fixed before riding the bike. Not sure how this will translate either but what you have there is NOT a Mary Jane Tinklepants situation.
Best of luck getting it fixed, and also good luck getting the factory to repair it. Not sure what Kawasaki's stance is going to be about this but it will be interesting to hear how it goes.
Brian
Trying to figure out what happened, new frames at the end of video.
http://youtu.be/TlLTAePregI
MGvalerio.