I just texted the ex to ask if lacquer would flake. He said that he thought any finish would flake.
If it's still flaking, you're going to need to sand it back to the point that the finish is in good condition.
I have a book that Miriam recommended. I think I bought it used from Amazon. It's called The Furniture Doctor by George Grotz. It was written in 1962. It has lots of detail on refinishing/repairing/refinishing. You many be able to get it from your library. It's also available both new and used from Amazon.
Best guess - it's lacquer. It dries quickly, so it was (and may still be) used extensively in factories for that reason.
First, I would rule out shellac. You can test for it, as shellac is alcohol solvable. I understand that you can wet the finish in an inconspicuous place with an swab dipped in alcohol and the finish will get sticky. (I tried this once, but couldn't tell if it was sticky or not.) If it tests positive for shellac, you can get a tiny can of shellac at home depot (or any paint store). If I remember correctly, it comes in clear and amber colored. I'd try the amber color. I would try a small spot on the edge of the marred area and see if it comes close to the color of the rest of the machine the next day. If it does, you have your answer. If not, you're back to trying something else.
Lacquer has to be applied in light coats, which you can very well do. You have to let it dry and then spray again. What I don't know is if the lacquer will darken when it dries to make the bald spot match the rest of it.
Several people have lots more experience than I do. If you get someone with actual experience to help you, listen to them. My experience is second hand, at best.
bkay