my guess is that you are losing spark on the rear cylinder. you could try removing the small black wire that connects the 2 coils. if it runs fine without that wire you need to replace that. that wire has a diode in it that can cause problems. note that the engine will not shut off with the key when that wire is removed unless you have a functioning antibackfire solenoid. you will need to use the choke to kill it.
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.