I have LT-1000 from early 2005, purchased new, Model 917.275371, 42" deck.
Last week I pulled its battery off the 2-amp trickle charger where it had been since last usage in the fall, and installed that into my lawn tractor.  I sprayed in some starting fluid in, it started right up, and I started mowing the weeds (some almost knee high) in my 1.5 acre "pasture".  After about 1/2 hour, I stalled it on some thick weeds, cleared out the clog in the deck, mowed some more.  After it stalled out again on more weeds about 15 minutes later, I tried to re-start and only got a faint click.  So I pushed it into my garage and put it on my Harbor Freight lawn tractor jack (a great tool/value, by the way). 
I went under and pulled off the drive belt and the mowing belt by hand at the crankshaft pulleys and let them hang free so I could be certain I was dealing only with the engine.  I put a weight on the seat to actuate the safety switch (just assumed that witch was OK). Since I still got nothing turning the key while pressing down the clutch pedal, I pulled out the wiring diagram and manual, I unbolted the gas tank and moved that aside, checked the 20-amp fuse (OK). I was getting power to the "B" contact on the ignition switch connector, and I checked the starting circuit of the ignition switch itself with the ohmmeter while the harness was disconnected, all good. I was getting power to the solenoid relay but not out the large red wire to the starter/alternator.  So I was stumped, figured that the solenoid/relay wasn't actuating, tried a jumper wire from the hot terminal of the solenoid/relay to the starter connector itself, still nothing.
After dinner, decided to take another look-see. this time measured 6 volts at the solenoid/relay connector.  So I went back and tested voltage at the battery itself, was 12.5 so I figured I had a bad connection somewhere.  So I took off the battery connections and re-introduced them to Mr. Steel Wool, and cleaned them well, and then all was working.  So my mistake was that since it had started up immediately after re-installing the battery, I had neglected to take a minute to clean up the battery's terminals and the lawn tractor battery terminals.  So just a note to us all.
 
Now a question: I noticed in the "recess" just forward of the ignition switch/solenoid area an insulated female connector and an insulated male connector not attached to ANYTHING, just hanging loose, see pictures.  So I just left them as they are, does anyone know the function of these?  Thanks.