old single handle shower handle loose

We have a basement shower with an original old single lever handle that is loose and not staying in place.... ie: it will slip down or over and the shower will go alternatively too hot or too cold. It also is tricky to get it to shut off completely... gotta jiggle it a bit into just the right spot. I would like to repair/replace it and am looking for advice as to how to proceed. I've replaced double handle set-ups before but never the single handle type. This looks to be a "ball" and not a cartridge type. I haven't taken off the back plate yet as the screws seem to be stuck and I didn't want to be hasty and break something.... we're still using the shower.

I've posted some pictures. The first pic is sideways... guess they don't do portrait orientation pics on this site (?). The handle is actually in an up vertical position. The last picture shows a screw (sorry, blurry pic) on the bottom of the handle that I tried to tighten, but it wasn't going anywhere.

Any repair I can do? Or will I need to replace it? IF so, thoughts on the difficulty of finding something that would be able to use the current plumbing set-up? (I realize that to answer the last you may need me to get that back plate off... just let me know)

Thank you for any advice!

WOW!! 30 yrs doing service and repair and I have never seen that one before.
If you do find out what it is can you come back here and let us know.
Nice pictures though. I'll see what I can find. Are you in the USA or in Europe?
helps to know what region of the world you are as well.

Hi Mr. David,

Thanks for replying and any help you can give! I live in the Pacific NW USA. The House was built in the early 60's so I'm guessing the handle is original from that time period...