Pump motor problem

I wonder if anyone has ever seen anything like this one:

The half-horse capacitor-start motor on my shallow well pump won't start reliably. What I've found is that when the motor shaft is in a certain position, it won't start, just hums. I can get it to start by rotating the shaft about 30 degrees, off of this dead zone.

The shaft rotates freely all the way around, nothing is binding. The pump runs and sounds normal when it does start. The starting contacts are pretty pitted, but it always starts fine if the shaft isn't in the bad spot.

I was hoping it was the starting cap, but I suspect it might be one of the motor windings.

Thanks

Phil in Minn