RE LK 3504 (35 hp 4 wheel drive Tractor starts, runs 5=10 minutes under load, then slowly fades away running slower and slower until it stops. Restarts in 15 min, same result

Sounds like you are starved for fuel. Look for a pinched off fuel line or a plugged up fuel filter. That would be my first place to start. Possibly even a bad petcock on the tank.

Check wheel balance.

plugged catallic convertor?

The battery is run down too low. When the light came on, alternator stopped working- the vehicle ran off the battery, until the lights and other electricals ran the battery down too low.
Recharge the battery, and have the alternator tested. It will probably need replacing. If you bring it into a parts store, they will test it for free.

It sounds like your alternator might be starting to go bad and the Battery Light is coming on because it is not charging the battery correctly. Have your Alternator checked with a volt meter, it should be reading 13 to 14 volts if working correctly. If not, replace your alternator.

There is a vacuum switch on the transfer case left side,up high!(Most have 2-3 vacuum hoses) That switch is bad!

You got a clean air filter? If it can't breathe, it won't run well or at all. Just like you. The car run okay until it's under load (driving)? If so, the fuel filter may be clogging. This will get worse till it won't hardly run due to no fuel. Fuel pump failing? The two cars I've dealt with that had clogged filters, would run less and less well under load, acclerate slower and slower and finally just die or not start. GM car, hmmmmmm. Fuel filter may be in engine compartment. I'm not sure on placement. My Tahoe's is on a frame rail but I think you have a unibody vehicle. If that's so, probably in engine compartment or it's tucked away somewhere. It's a canister looking thing with an in and an out for fuel. That's it. Not complicated just depends on placement as to how hard it's gonna be to access it. Fuel pump is a different story. In the gas tank, tank needs to be removed and pump replaced. Hope this helps.

sounds more like the spider gears in the front end !!!!!!!!

have your catalitic converter checked

Sounds very possible but it also sounds very expensive for a simple fuel pump? He must be on a good rate. In Australia it would cost about 4-5 hundred for that repair at MOST and the US dollar is nearly worth twice ours! *Shock*