The previous posts seem to indicate that "Captainscall" discovered a bad cell in the battery. He also cycled through a fair number of voltage regulators.....but having the bad cell could have been the culprit and MAYBE not any of the voltage regulators.....the post was convoluted.
Do a simple test....connect a test meter to the battery terminals and watch the voltage with the ski running...should be greater than 13 volts. If the battery voltage is lower and / or jumping around could be a few things: (but it means the ski is not refreshing the battery with DC voltage)
Starting from the back of the ski to the front:
1) Check top connection to rear electric box next to battery, 6 pin connector. check for any minute signs of corrosion on pins and legs, clean it up if suspect.
2)Inside the box is the solenoid and the ignition coil, make sure all connections are clean and bright and tight
3) should have Red/purple wire though a 15 amp fuse from top connector to the RED solenoid post (this is the line where the ski gets the voltage back from the regulator to keep the battery refreshed and above 12.4+ volts. Pull fuse, check, re-seat or replace.
4) check continuity of Red/purple wire from 6 pin connector all the way to the front electric box in the nose of the ski to Red/purple coming out of the ECM...pull and re-seat all harness connections along the way if continuity is not there...clean connections if necessary/appropriate if continuity is restored just by reseating connections
5) Check 3 yellow wires (inside front electric box) from stator, should have .6 ohms across all connections 1-2, 1-3, 2-3 (or close, fairly equal)...this tells you stator is probably OK.
6) Check 5 amp and 15 amp fuses on the ECM, pull and re-seat if OK
7) with ski running....should have 13V + on the RED wire out of the volatge regulator feeding the Red/purple wire back to the solenoid (and battery)
8) Opps forgot....in rear box check wire connections to top of solenoid posts and check for good ground connections to the ignition coil
No cost to check any of this stuff...just patience.
I'm getting the distinct impression that the older skis suffer more electric and carb problems than major mechanical issues....that has been my personal experience so far.....the hardware seems solid....everything else is flaky (but age does take its toll).
Happy hunting...good luck.