Sounds like a great rideout, the GTR will always attract attention, maybe that's why we all get so hooked on them
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With regards to the shift problem, could be a number of things but because you can get the lower gears by jiggling the pedal there are two things that 'spring' to mind.
1) Change shaft return spring not centring properly. (should affect all gears)
2) More likely, would be a sticking ratchet pawl or pole spring on the right hand end of the shift drum. Make sure the stopper plate goes back the right way round.
Could also be something has come loose / broken on the shift mechanism but my money's on the springs. Bad news, both right side covers, clutch, carb, oil pump have to come off to get at them. While it's stripped and before the side cover goes on, check you can get all gears by manually turning the shift drum. This will show if you have a sticking/bent 3rd gear shift fork, that would be a complete strip down. Not a 5minute fix I'm afraid.
Try to keep the clutch plates in order when removed and if you can, make sure they go back in the same slot position. It will be interesting to see if you have any broken plates after those 3rd gear starts
Brian.