This happens to me too sometimes, just either come to a stop in second then shift down, or let the clutch out then pull back in and it should go down. Sometimes you have to re-engage / disengage the clutch once after stopping as well to get it into first. Just get used to it and anticipate that it might happen so you dont get surprized and dump the bike or something.
it sounds like it might be your shift drum. kawi's are notorious for the shift drum wearing, usually its from first to second, but either way it sounds like you will need a new shift drum and maybe a shift fork as well. my roomate pulled his tranny apart and that is what happened to his and i guess it happens alot to kawi's. ill see if i can find the link to the how too and show you whay im talking about, or maybe ill take some pics of his. but from what i've heard its a fairly common occurance.
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