I've been in the process of rebuilding my B&S 31C707-0154-E1. I can't remember and can't find where it goes. The part number for the wiring harness is #698329. The wire runs under the front of the engine block plugs into the solenoid on the bottom of the carb with a pigtail. One wire then comes off the pigtail and end with an eyelet. I think it's for the engine shut off, but I can'f find where it would hook up. I haven't been able to find anything in the service manual or online.
Sounds like that is the ground for the fuel solenoid on the carb. Look behind the carb solenoid and there is a bolt with a 5/16 head holding the sheet metal to the engine cylinder. That is were the eyelet goes.
