Cub Cadet Question

My old 1250 Hydrostatic Cub is having some electrical problems and I am trying to figure out if it is the alternator or battery (or maybe something else).
Observations: - I can successfully jump start the tractor with my truck, and it seems to run just fine, however, the Discharge/Charge meter is showing slightly Discharge.
- When I turn on the PTO, it runs for a little while (few minutes), then just dies. If I turn off the PTO, it typically comes back to life, but turning on PTO just causes it to shutdown. PTO will run again after a jump start.
- I took off the right (sitting in the seat) cover panel to reveal what is labeled "Alternator Electrical System." I assume this is the alternator. Anyway, when I jump started the tractor, this device moved, but the entire time the tractor was running (attempting PTO again and jogging throttle) the device did not appear to move. Of course, all I can see of it is the rubber part that you can rotate with your hand (maybe something internal is moving?).
- While the tractor is running, the voltage across the battery terminals is around 8V.
- The tractor will not continue to run if the battery is disconnected.
- I put an ohmmeter on the battery and did not read any resistance under 20 KOhms (I was possibly trying to see if the plates were shorted).
- The battery voltage (no load) was reading about 11.8V, but as soon as I turned the key, it dropped to 4V.
Questions: - Is the alternator running and I can't tell? - Is the alternator always supposed to run? - Could a bad battery cause it to not run? - Is the alternator bad? - Is the battery bad?
Thanks in advance-
Chris
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Clarification:
The alternator seems fine, but it is not engaging the main gear? For whatever reason, it does not extend.
Any ideas?
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