Coming out from under the top sheetmetal somewhere on that engine, you'll have a pair of wires. One red and one black. They'll terminate at a white plastic plug, and from there 2 wires continue on.
The black wire is AC voltage, for powering the headlights. With the plastic plug unplugged from the tractor's harness, you can test for AC voltage output with the engine at speed. It'll produce quite a bit, like 15+ volts AC, maybe even up to 30v AC.
The red wire is DC voltage that charges the battery. Just prior to the plastic plug that both wires go to, you'll see a black heatshrink protected lump. This is the charging system diode. This keeps the battery from draining backward into the stator windings when the tractor is parked. If this diode is blown, the battery will drain during storage. Having a battery under a constant drain will ruin the battery and the result is like you see when you go to start, the battery voltage drops off rapidly to 6v.
You can test for DC voltage at the red wire, just like the test for AC voltage at the black wire. You should see 13-something and it'll increase with engine speed....maybe as much as 15v.
These systems are "sensative" to battery voltage, so if you have a really drained battery, or a bad battery, and start the tractor by jumping off a car battery or jump box, the system voltage with everything hooked up, might be really low. The stator is only 2-4amps of charging power, so that's really low. In other words, if you test charging system performance, make sure the battery is fully charged, otherwise, you'll get a false low reading and be chasing your tail thinking you have a bad stator or bad external voltage regulator(if equiped).
The final test is of that diode. You'll need a diode test function on your electric test meter in order to do this test. The way I do it is to cut off the heat shrink to expose the diodes bare metal wire leads. I put my black test lead on the engine side of the diode, and the red test lead on the plastic plug side of the diode. In other words, will current flow toward the stator. It should not, and the meter should show "open loop" or "no continuity." Then I reverse the leads, and the meter should show "closed loop" or "good continuity."
If you find the diode is bad, you can purchase just a short "pig tail" repair harness that contains the diode and the plastic plug-in. Available on ebay and other sources for $14ish. Simply cut off your existing red and black wires just on the engine side of the diode, and splice in the new pig tail. 5min repair.....and it'll save you a $60+ stator that takes pulling the flywheel to replace.
Hope this helps.