I just found a tag that shows it was made in Des Moines, Iowa. The model # starts with 536210 the rest is unreadable. The serial # is 6-49682. These are the current pics of the rebuild.
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The more I look at that one, the more it looks like it came from the same people as made these. These are Murray but the underlying designs I believe came from some of the companies that Murray acquired, including AYP and Roper (I think)
Though the paddles are different and the 3d stage housing is shaped alittle different in the curve of the addon 3d stage housing, there's alot of similarities between yours and this one. Additionally the base auger housing has the same shape (Craftsman 536.918900). This one is a gear drive transmission unit.
This is where the similarities really come into play. This one has has a friction disc drive but the auger gearbox, augers and base housing look like yours. This is a Craftsman 536.90515 machine
Additionally yours also looks like it has the bolton auger bearings on the end of the auger housing, I suspect the shafts only run half way down the augers and the ends of the auger housing bolt on like all the Murray originated ones I have.
I suspect that the basic design came from something like AYP but may have been sold under either label, assuming it was not a Craftsman labeled machine.
I don't know how Murray assigned serial numbers to their own labeled machines, but those sold through Sears started 536 so that may be your starting point.
Good luck.