[bestgenman]

First, tell me which generator manufacturer we are dealing with.  That will help.  

If the engine will stay running with the "stop solenoid" manually tied or held with the plunger full embedded and will not stay running if you release it , then your trouble is electrical.  Something in a series of switches is not making contact or a wiring issue (like the terminal story above).  This is where the brand is important as wiring schemes differ between makers.

Tell me if there are two or three wires on your stop solenoid.   As posted prior, the black (NEG) and red (IGN) are the hold in functions.  If you put a light or meter on these leads, does the light stay on while you hold the plunger in?