While the engine is cool, it works fine, with no problems starting it and normal power output. Once the engine warms up, it starts to stutter and die, especially during variantions in power demand as in turning from uphill mowing to downhill (and vice versa!) or engaging the mower deck. When it starts to stutter, sometimes one can "rescue" it by engaging the choke, but not always. After it died, it can be re-started with 5-10 seconds of starter applied.
After the engine developed the problem my first reaction was to checkintake the fuel line, fuel filter, air filter and fuel pump. All of those were Ok, though. So I took the mower to the mechanic who re-checked the same things and also found them to be Ok. He also checked the carburetor with no problems found. The mechanic was advised by John Deere to replace the ignition coils (the motor uses two separate coils for its two cylinders), which was done, but without success.
Since the mechanic seems to have run out of ideas, I started to fiddle with the engine a bit. One thing I noticed is that, with the air filter off and the carburetor intake in plain sight, one can see a fine mist of fuel being ejected from it while the engine stutters. So there seems to be air travel in the wrong direction, at least during some parts of the cycle. Something else that I noticed is that the intake manifold (part A in the second photo on Page 70 in the linked PDF) is metallic clean on the end for one cylinder but kind of sooty black on the other cylinder's end.
From those observations I've deduced a working hypothesis, that something about the intake valve on the "sooty" cylinder might be wrong sending some exhaust into the intake during some part of the cycle. I've adjusted the valve clearance as explained on page 32 to no avail. The next step would be to disassebmle the cylinder head and inspect the valves themselves. Either related to improper sealing of the valve or valve timing. This, however, is somewhat outside my comfort zone, as I am just a computer software guy. Can someone with more gasoline running in their veins maybe advise me on how to proceed and whether my hypothesis makes sense or I am barking up the wrong tree?