Based on what is posted so far I have to believe that this is a fuel issue. I understand the sled has spark even after it dies so I want to rule out bad coils. But just to be sure, how did you determine that it has spark as it dies?
So on to fuel: When you start the sled do you need or use the primer? And, after the sled dies, if you use the primer can it be restarted? I am asking these questions to try to eliminate a bad or blocked fuel tank vent.
How much of the sleds history do you have? Have you thoroughly checked the carburetors? Do they have the correct jets? Are the power jet lines connected correctly?
Have the crank seals been changed?
There is a question about fuel mix ratio. Has the oil pump been disconnected? And if so, can you tell why?
We have a number of sleds that have come in with the same story "ran last year, just needs carburetor adjustment." But the story is always the same and it becomes obvious quickly that the sled has not run for several years. First crank seal failure causes a piston failure, the owner assumed the piston failure was caused by no oil and a failed oil pump so they disconnect the oil pump to run pre-mix only to have another piston failure, at that point they have run out of talent and park the sled. We have probably had six or eight Intruders all come in with that same story. Nothing wrong with the oil pump. We clean it up, put in fresh crank seals and fresh pistons and away we go. I am not saying all sleds are like this, but if the crank seals have not been checked, quite a bit of the other troubleshooting may be meaningless.