The technical term is genset - generator set.
The actual generator is only half the genset - the other half is the engine. I'm not trying to nitpick, but I have to keep terminology correct, as it prevents a lot of misconfusion in testing.
This sounds like you probably have a breakdown in the ignition system. so lets start there.
Run the genset until it shuts down. hen it does, immediately check to see if you are getting spark to the plug.
If you are, let me know.
If not, remove the spark plug wire from the plug.
Remove the small white wire from the coil.
Set your meter to ohms, and place one lead to the ground wire of the coil, and the other to the terminal you removed the white wire from. You should get 0.5 ohms.
Now place one lead to the ground wire of the coil, and the other lead into the spark plug wire. You should get 1,100 ohms.
If either reading are out, the coil is bad. Remember, the coil must be hot when testing, so if you let it sit too long and cool down, you may have to start the genset back up.
If the coil tests good, set your meter to ACV.
Connect one lead to a good engine ground. Connect the other lead to the white wire you removed from the coil.
Press and hold the start button for 3 or 4 seconds. It should read between 15 and 50 VAC. If it does not, the magneto is bad.
Let me know what you find.