hawk6, Welcome to the fold. After years of playing with these machines and sharing the interest, I consider this to be THE Hawk community, glad you could join us. If you have the '04, check to see if it still has a plastic airbox or if it has pod filters. MUST have pods and a snow flap to run and keep the motor cool and properly carburated to stay alive. If you don't have pods, let me know, I *might*
know where a set are. The other thing to determine is if the pistons are original or if they have the "upgraded" piston rings. According to an unnamed source at Boivin, the original 2004 600 factory Doo motors had almost a 25% failure rate because of the flawed chrome facing on the rings. The chrome flaked off and scored the cylinders, killing pistons and cyls in the process. Expensive to fix after, cheaper to fix before, it happens. Maybe you wont be part of the 25% and have the problem. Fingers crossed. Doo had a similar problem with their '01 800 top ends and replaced them all as a recall. Not enough failures with the 600 to justify recall, so it was handled on a case by case basis. Not trying to alarm you just an update on the '04 model quirks... Plenty of guys running original '04 600 motors without problems.
In addition to the gusset that TCSKI mentioned, the exhaust joint that IB mentions is where the J-pipe is welded to the muffler its the one prone to failure, had to reweld a few over the years. Silicone the pipe springs at the muffler joint also, I seem to remember a spring tab or two pulling off from vibration.
Got any other Hawkers in your area to ride with?
Also check the muffler on your hawk they will start to crack out and performance will fall off. It is a easy fix but one to keep an eye on. Good luck and enjoy.