The biggest reason that these do not start is because of a dirty carburetor. Many people do not know that you CANNOT store ANY engine with gas in it for longer than 30 days....EVEN IF YOU USE FUEL STABILIZER in the gas. This causes gum and varnish to form inside the carburetor and restricts or plugs the very small passages.
Next could be a bad electronic ignition. Check the spark with a good spark tester and make sure the ignition is working. Keep in mind that ignitions will sometimes work part of the time and not work other times., But first just check for spark. If you have spark, go on to the carburetor and other items I will tell you about.
Next, These engines have a history of needing the valves adjusted. The MUSY be adjusted about every two years or more often. The clearances should be between .004" and .005" between the valve stems and the rocker arms when the valves are completely closed.
Next. Check the flywheel key. These engines also have a history of shearing the flywheel key. This causes the ignition to be out of time.
Since you have said it has good compression, I will take your word for that, but make sure that it is even on both cylinders.
I now go back to the first and most likely cause, the carburetor and fuel system.
Check the fuel delivery by removing the fuel line from the carburetor and turning the engine over as if to try to start it. While holding the end of fuel line over a coffee can, check to make sure that you are getting a good flow of fuel out of the line as the fuel pump is pumping. If good fuel is flowing, precede to the carburetor. If not you may have either a bad fuel pump or a plugged fuel filter.
Here is my comprehensive carburetor and fuel answer that may give you some ideas.
As engines sit or get older, fuel that is left in the carburetor can turn to gum and varnish and cause this and other problems. Also, any gasoline that was left in a gas can for a period of more than 30 days must be discarded because it also has begun to turn to varnish.
Today's gasoline's contain MTBE and alcohol. (Ethanol) They turn to "Junk and garbage" very quickly. Alcohol is partially water (H2O). And they call it "Oxygenated fuels! It is the oxygen that breaks down the organic compounds in the fuel and turns the gas to "Garbage" (Gum and varnish) The fuels we had just a few years ago had no alcohol in it and would store for longer periods of time before going stale... and fuel stabilizers do almost nothing to prevent the fuel from going bad with the changes in today's fuels.
Do not buy gas from the "Discount" Stations. The discount stations get a reduced price on gas because they may be buying fuel that is nearly 30 days old already. You may be getting fuel that's nearly stale right from the pump when buying from a discount station. Purchase your fuel from the well-known stations such as Shell, BP, Sonoco, Phillips 66 etc.
More than 70% of all of our repairs in our lawn mower business are due to this same issue. You most likely have dirt, gum, varnish...etc in your carburetor plugging up the small passageways and jets in the carburetor.
The carburetor must be removed from the engine. Clean all parts with carburetor cleaner and blow out all the small holes and passageways with compressed air. Remove all of the non-metallic parts since the carburetor cleaner will cause them to be disfigured decompose and plug the carburetor as time goes on.
Wash the carburetor cleaner off of the metal parts by washing them in warm, soapy water then rinsing with clean water. Dry them by blowing it off with compressed air. Make sure that all the passageways are blown dry before reassembling.
Reassemble using a NEW carburetor rebuild kit.
DO NOT TRY TO REASSEMBLE WITHOUT USING A COMPLTE CARBURETOR KIT! You will just end up having to do the job again.
ALWAYS clean the fuel tank and replace the fuel line when doing this repair or you may have to do it all over again. The inside of the fuel line disintegrates over time and these small pieces of rubber will plug up the carburetor too. Dirt and water from a dirty fuel tank will also plug up the carburetor. Find the Model, type and serial or code numbers off of the engine and take them to your local dealer to get the carburetor repair kit.
If you do the work yourself, take pictures or at least make a drawing of where all the linkages, gaskets, and component parts go. Correct reassembly is critical.
If the carburetor still doesn't work correctly, you may have to take it to someone who has an "Ultra-sonic" cleaning machine. This machine uses carb cleaner and ultrasonic vibrations to get the very small passageways clean when traditional methods fail.
Here is where you can get an inexpensive "Ultrasonic Cleaning Machine"
If you don't feel comfortable with this kind of repair I would suggest sending it to a reputable shop.
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