Your cartridges are not pinfire cartridges, they are some bastardization of some other cartridge.
Here is a better pic of your gun, along with a couple of my pinfire boxes and some example of the 15mm cartridges.
Also, the most common name for this is a boxlock pistol.
I can't tell if yours is 15mm or 12mm without something for scale, but it is one of those.
