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Have you checked the spark plug wires at the coil?
You found a loose lead, I'm guessing you have 2 coils so you should be able to measure the resistance from each plug cap to chassis ~8,000 Ohms, it should be the same on both sides. Or is it a single?I googled it and came up with a Ford manual that certainly looks like a suzukihttps://www.manualslib.com/manual/1053929/Suzuki-Burgman-Uh125.html#manualIf you could reference the correct manual I could possibly be more help.
https://www.manualslib.com/manual/794402/Suzuki-An650.html
Thanks for posting the manual, it looks like it should be fairly straight forward looking at the schematic on page 174Make sure you have 12 Volts on the fuel pump relay coil which will also mean you have it on the coilsThe thing is the ECM must ground the other side of the pump relayI expect it might run the pump for a couple of seconds when you turn the key onThe Camshaft position sensor or the Crankshaft Position Sensor tell the plugs to fire and the pump to run.Do you see any spark or hear the pump as the motor turns over?I often suggest that a small lamp attached to the pump relay contact and chassis would be worth its weight in gold as a troubleshooting aid in fact its the first mod I made to my new V7One other thing you have is the Tip Over Sensor, I expect that could be disabling the ECMDoes the dash show any alarms?I'll wait to hear back before I add any moreGood Luck
You probably won't find 12 Volts at the pump, its only there while priming and later on when the engine is cranking.(Attach a light there if you can, you will see it flash on for a second when you turn the key On and should see it come back on while cranking, a light is much easier than using a meter)Ok you have spark when you turn it over, that tells me the position sensors are working and I suspect the Tip-Over is OKYou could try a shot of starting fluid just to ensure the sparks are firing correctly, it should run for a burst.So you need fuel, perhaps the pump is gummed up or maybe just the injectors are gummedPumps usually put out about 40 PSIMight be a hose off in the pump circuit also
Have you tried whacking in the general vicinity of the pump. I've started many a stalled GM pickup truck by whacking the bottom of the tank while someone was cranking it over. Known in the industry as the "GM Tap Test".
When you hit it with 12 Volts do you get a good healthy spark off the wire as you scratch it on to the battery?If not there might be varnish on the commutator from dried out gas this can insulate the brushes, this is quite common with a bike that's been sitting.My Bro Kiwi Kev got a flying brick going for me I will ask him to comment. If yes a spark but no whirr its jammed up Either way try to get some carb cleaner into the pump.