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I have a scag STT61V-35BVAC and the mower runs for awhile and then shuts off. I think it may be the fuel solenoid, but I cant find it. Would love some help.....

It resides on the bottom or side of the float bowl if that model has a carburettor engine
Easy way to eliminate it is to either power it from an external battery ( confirms the wiring ) or to replace it with a bolt & O ring or fiber washer .

Depends which the OP has as there three or four different engines and one of them is an EFI engine. So the engine info is needed.

Also which one of the seven STT61V-35BVAC is it?

engine info 613477-2186-J1
Solenoid part # 842109
still can't find it

That engine came with two different carbs available. One with a fuel solenoid and one without. If you have no electrical wire going to the carb, you have the one without a fuel solenoid.