Gx390's have 389cc displacement. Could be a sign for you there...
Carb looks funky. I have only seen that fuel shut off solenoid on the larger v-twins, pretty neat! Anything stamped on the engine block anywhere?
Also, gx340's and gx390's have a lot of interchangeable parts (tank, shroud, flywheel, pull start, carb, air filter, could have more I am only sure about those specifically) could be a gx340 clone based on your 337cc statement... No idea though, just guessing. How did you know it was 337cc?