Single Phase Puzzle to challenge you

[nicebutdim]

It has brushes?!

OP did state this was a puzzle.

 

[brianmoooore]

I've been asked to 'look at' a few of these DIY type compressors over the years, and the usual scenario is that one of the capacitors has failed. Replace it, and it'll run for a week, until the replacement capacitor fails as well, which is probably because the winding has developed a few shorted turns.
These tend to be cheap and nasty single phase motors, which aren't really single phase at all. They are modified three phase motors, using the run capacitor produce a phase change and simulate a poor three phase supply.

 

[DefyG]

Photo of the name plate might help

 

[Marvo]

Have you got the lid of the connection box on the motor? Often there's a wiring diagram inside the lid.

What is the make and model number of the motor? Somewhere on the motor there should be an information plate, can you post a pic of it please?