Sorry for the long post, trying to give as much information as possible.
2001 RX
Well, I pulled the starter and rebuilt it. Connected it back to the engine for a test and I still get the same response. The starter solenoid just clicks real fast. Here is the history of repair so far...
Replaced fuel lines and pulled carbs for cleaning including the fuel filter (badly needed it). I did this because the ski would run for 20 minutes and stall out. Put everything back together went to start the ski and it would not start, the solenoid just clicked. Replace the solenoid and the same thing happened. I then bridged the power cables to jump start the starter so to speak. That worked. It ran great for 40 minutes and it was sun down. 4 days after the run I went to have an Fast Lanyard programmed for it as I was missing mine. I just had a learners Lanyard. Got to the dealer and the ski had absolutely no power. I got power very briefly, long enough to have the lanyard programmed. However, it still would not start.
I brought it back home and chased the voltage around. I found that the 15amp fuse in the electrical box was extremely corroded and I was losing power going through the fuse. Replaced the fuse and cleaned the pigtail and terminals of the corrosion. Put everything back together and I now have power to the lanyard. I tried to start it and all I get now is the clicking noise form the Solenoid. So, I replaced it with a known working one and still get the clicking. I also rebuilt the starter at this point just thinking that this might be the issue as I earlier had to bridge it to get it to start.
At this point I have no idea what to look for. I don't understand why the solenoid just clicks. Even after rebuilding the starter, I can't get it start even from trying to bridge the power cables together. I've had this thing for 2 months and have drove it for 40 minutes! Please help!!!!