STX38 yellow deck won't start

The starter doesn't spin, however. Last week a repairman told me the starter was shot, so I had it rebuilt - now the starter should be good.

I had 12v between the starter's positive terminal and the frame, so that sounds good. Thinking the cable from the battery was hosed, I jumpered the battery directly to the starter as an experiment. No change.

Physically, this thing is a heavy cylinder with a gear on top (the starter?) coupled with a smaller piece (the solenoid?). The battery runs directly to a post on the solenoid. Another wire runs from the key to the solenoid also. This wire has a much smaller diameter than the power cable so it can't be used as a ground I am sure.

Curiously, the starter part (not the solenoid) has a rubber sleeve (for lack of a better description) that looks like a wire is supposed to run into it. There is no wire anywhere, however, that looks like it needs a home. I think it's a holdover from a design change, but I'm not sure.

So what could be wrong? What else is left to check? Battery - wire - solenoid...
My starter has no explicit ground on it, by the way. I'd expect to see a ground as heavy as the red power cable. Does it ground to the engine block?

However, over the winter the key was left in the 'on' position which drained it as flat as possible. It seems to be holding a charge however.

I'm actually kind of confused regarding the difference between tractor and car batteries. How do I know that the car battery won't damage the mower?

I shorted the starter to the 'key' wire, and the starter engaged and spun the engine a few times. Then the battery was done. Odd, given how I thought it had a full charge.

Then I jumped the mower with my truck. The mower sputtered once, blowing a winter's worth of crud out of the carb. It fired right up on the 2nd try. I removed the jumper cables immediately and took the truck back to the front of the house, leaving the mower running.

About 5 minutes later, I made it back to the mower. In the process of getting ready for a test cut, the engine died. Having nothing to lose, I turned the key, and it fired right up.

I am at a loss to explain what is happening. The battery can not charge in 5 minutes.

Now the problem is that after mowing for a while, the mower will simply stop running and won't restart for a while. I installed a new ignition coil, so a lack of spark shouldn't be the issue.

I also think the engine is running too rich. On my old Mustang, the carb had a pair of needle valves that controled the fuel/air mix. Does the mower have something similar? I looked and only found an idle setting.

The only other problem I found was that a tube leading from the carb to the front of the mower, near the spark plug, had rotted and cracked. In the spirit of backyard mechanic hacks everywhere, I wrapped it in black tape so I could get on with it. Maybe it is wishful thinking, but the mower seemed better afterwards. I don't know what the tube is for, looks like a vacuum line. It isn't a fuel line.

Now the problem is that after mowing for a while, the mower will simply stop running and won't restart for a while. I installed a new ignition coil, so a lack of spark shouldn't be the issue.

I also think the engine is running too rich. On my old Mustang, the carb had a pair of needle valves that controled the fuel/air mix. Does the mower have something similar? I looked and only found an idle setting.

The only other problem I found was that a tube leading from the carb to the front of the mower, near the spark plug, had rotted and cracked. In the spirit of backyard mechanic hacks everywhere, I wrapped it in black tape so I could get on with it. Maybe it is wishful thinking, but the mower seemed better afterwards. I don't know what the tube is for, looks like a vacuum line. It isn't a fuel line.

This thing hasn't been tuned in years. Condenser and points are years old, maybe as old as the mower, which is 12 years.

Maybe it's time, eh?