Sad to report you have Mixet brand. It was never anyone's favorite brand, and they have been out of business for a decade or more. The good news is , almost any hardware store has your cartridge and replacement handles. Take your old cartrigde with you, as there are a couple of variations.
If you do no have a recirculating pump on your hot water, then you will not be bothered by some classic internal defects in the body, and you can probably keep the shower going for a few more years.
Nomix Inc. repair cartridges that fit Mixet brand valves
It is a Moen style stem to replace the Mickey Mouse one that Mixet uses.