Bolens Mower Won't Start?

I have a 2003 Bolens riding mower, model #13AN683G163 which will not start. This began over a year ago, when I thought it was the battery. I replaced the battery and the mower started, but the battery would drain and have to be charged to be used again.

This year, it would not start even after charging. I replaced the solenoid, the spark plug, and the gas and air filters. The fuse under the seat is OK, and the gasoline is new. I have cleaned and lubricated what I believe is the starter, which is on the side of the engine, and often the mower does start.

Now, however, when I turn on the key, the flywheel turns a little, but the mower does not start. I have checked schematics online and even called Bolens, but I cannot find a starter, and Bolens has told me that there is no part called the starter. Any ideas?