When you say it shifts to 1st then back to neutral do you mean that it engages into first gear but then pops out of gear? Or do you mean you push down and it goes into first but then not another gear? Just trying to see what you are dealing with. If you can tap it down into first and the shifter returns up and it stays in gear but then you can't repeat this and shift into second then my guess is you have a bad shift stop, could be related to the shifter being welded on, someone might have been shifting it really hard and stripped out the shifter so they welded it on, and in the process of continuing to shift like a caveman they messed up the stopper.
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....