Welcome to the forum achtung!
It would appear that your tractor has an electro-magnetic PTO clutch which explains why it shuts down when the voltage in the battery drops too far. The first thing that I would recommend is that you charge the battery fully, test the voltage, then start the tractor and test the battery voltage while it is running. The voltage on the battery should be about 12.5 directly after it has been charged and about 13 - 13.5 while the engine is running(I am supposing that you have a 12 volt battery). If the voltage is not higher while it is running it is a sure sign that the alternator is the culprit. Otherwise, you may need to replace your battery.
I'm afraid I can't help you with info on the engine. There should be model, type and serial numbers on the top shroud somewhere. If you find the numbers you can enter them into the Briggs & Stratton website below and find all the info you need.
http://www.briggsandstratton.com/support/need-help-finding-model-number/riding-mower-engines/
It would appear that your tractor has an electro-magnetic PTO clutch which explains why it shuts down when the voltage in the battery drops too far. The first thing that I would recommend is that you charge the battery fully, test the voltage, then start the tractor and test the battery voltage while it is running. The voltage on the battery should be about 12.5 directly after it has been charged and about 13 - 13.5 while the engine is running(I am supposing that you have a 12 volt battery). If the voltage is not higher while it is running it is a sure sign that the alternator is the culprit. Otherwise, you may need to replace your battery.
I'm afraid I can't help you with info on the engine. There should be model, type and serial numbers on the top shroud somewhere. If you find the numbers you can enter them into the Briggs & Stratton website below and find all the info you need.
http://www.briggsandstratton.com/support/need-help-finding-model-number/riding-mower-engines/