01 Honda TRX350FM no neutral light,reverse light staying on.no st

https://www.manualslib.com/manual/804553/Honda-Trx350tm-Rancher.html?page=294&term=fuse&selected=7
Wiring diagram in manual below.
https://www.manualslib.com/products/Honda-Trx350fm-3239929.html
https://www.youtube.com/watch?v=aRhAQLWhwJo
https://www.youtube.com/watch?v=flb6uUwty8c

It sounds like you need to adjust the shift lever or move it in an out of gear while holding the starter switch on. this will tell you it's the lever not the switch.

there are three sensors on the left side just in front of the rear tire probably a bad connection

check for faulty safety micro switches on the machine
(seat ,brake ,clutch gear position -neutral etc)

btm is neutral .all the way down i believe. all up from neutral position pull brake lever and push button to put in reverse ? from neutral pos. reverse is past neutral only gear lower than neutral is reverse

The neutral lights is not going out until the atv goes into gear. Look over the shift levers and rods that puts the vehicle into gear. Take the shift linkage loose from the engine and see if moves by itself. If all the linkages move except when attached to the engine, then the linkages could be out of adjustment (if it has multiple linkages like 4 drive) or the engine has some internal issues like trans problems or other related to the shifting mechanisms.

Look for broken wires on bottem right side of engine on park /neutral safety switch wires pulled off or broke

check for spark

does the neutral light come on? checked the fuse? it wont crank if its not in neutral and the kill switch is on. if thats all ok, test the solenoid.

remember on early models there was a safety pin that had to be held to switch to reverse,,,your 500 does this automatically so check for safety pin hookup. Could be brush or
grass jammed in where it moves.