Questionchain has inconsistant tightness (loose, tight) when rear wheel is rotated. Noticed sprocket has clockwise and counterclockwise movement when rear wheel is held fast.
AnswerBill, you didn't give a year, so I had to make a guess...
http://www.bikebandit.com/houseofmotorcycles/kawasaki-motorcycle-kz440-standard-...
No photos for the rear hub, but the parts list shows a rubber hub damper part number, which is pretty normal for many bikes. Your bike is over 20 years old and if it has a lot of miles, the rubber dampers in the hub are worn, damaged or shrunken down, allowing excess sprocket lash, back and forth when you put throttle on and off. The dampers protect the drive line from the shock of on/off throttle torque.
The condition of your chain indicates that it is worn out, as well. If the hub dampers are still available, order up a set, to install when you change the drive chain and/or sprockets.
Bill MrHonda Silver
