Are you saying the shiny part of the finish is flaking off or are you saying the veneer (a thin piece of wood on the surface) is flaking off?
From what I can see, it looks like it's just the finish. I'm not sure if the finish is shellac or lacquer. (I think Glenn covers that in his finishing tutorial.) I think the way to correct this is to reapply the finish that is missing from that area. Then you would proceed with the rest of the work. If it's shellac, it would be easy. You can do the french polish method that Glenn recommends in his tutorial on redoing the machine head itself. It would take a few days, but would not be too messy or require massive daily cleanup as you just use a cloth to apply the shellac and can dispose of it daily until you build up the finish to get close to the the other areas.
If it's lacquer, it's more difficult for the lay person. You have to spray lacquer. You can buy it in a can, though. Whether it will require staining to match before you spray, I can't tell you. I've been around people applying finishes, but I've haven't done that much myself (my ex owned a woodshop).
I wish I could help more.
bkay