Perhaps the thread at the link below will help you in a few ways.
Your trans may have locked up in similar fashion to the one in the thread, and for the same reason.
You will have to determine if in fact the bolt described in the thread has also failed on your tractor.
Plus, you may gain a better idea of what lube is available to you.

Here is a link that might be useful: Holey Cow MTD!

Thanks for that, some useful info there. Luckily, mine isn't that badly damaged - if it was I'd just be looking for a new axle!
The diff unit is fine - even when the axle was locked I could still turn each wheel (the other wheel would move the other way), suggesting no problem with the diff.
I suspect a bit of grit may have lodged in either the input or intermediate gears, locking it up. By the time I'd removed the axle, it was operating as it should be, suggesting whatever locked it had dislodged.
Apart from a few small gouges here and there (probably from the bits of casing), everything seems in good order, no missing teeth or anything. It's a bit of a mystery - I'm not even sure at this stage whether whatever caused the hole came from inside or outside the casing. I do know, however, that this isn't a new hole - judging from the state of the metal (and the grass inside the casing!) it's been there for some time.
I think my best bet here is to JB weld the hole, give everything a good clean, regrease it, put it back together, and hope for the best. If it still fails, I'm out an hour or so labour plus a tiny bit of JB. Worth a shot!