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A question first: what kind of charger, and what charge rate (Amps)? Most mc batteries are around 14-16Ah, and shouldn't be charged above about 1.6A (Ah rating divided by 10), or may be damaged.

At less than 12.6V resting the battery is not fully charged; I'm guessing it is time for a new one. The Voltage while running are a problem, and indicate a problem with the charge circuit, or a battery that is really failing after the charge. In any case, the battery should be replaced, then the running Voltage tested again. Make sure the battery cables are clean an tight after replacing it.
 

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A charger that works fine for vehicles with larger batteries can still over-charge a bike battery, A 'trickle' is a relative term, too; a trickle for a car battery can be a flood to a bike battery. A battery damaged by a charger can develop an internal leak, draining it over several days. You can probably look up the charger on line, and find out if it is suitable for a small battery.

In any case, your testing indicates it is now bad, and you need a good battery to diagnose a failing charging system.
 

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Take Voltage readings: 1)key off, 2)key on, not running 3)running at idle 4)rpm above idle; if you have a tach, about 3000rpm
Those readings will determine if the charging system is working, and your battery failed, or if the charging system killed the battery.
 

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Those numbers tell me the battery is probably in pretty good shape, since it bounced back after discharge. I think you need to go through the charging system now, checking all connectors from the stator to the rectifier/regulator, and from the r/r to the battery. Clean any contacts as you go. Also, with the stator unplugged, measure its contacts to each other and to the engine case; you are checking to see that the resistance is really low between each contact, and really high/open to the case. BTW, many meters will measure the contact-to-contact resistance as near 0, as measuring an Ohm or two is unreliable in them. There is a test for the stator output with the engine running, but, without a service manual, I can't tell what AC Voltage to expect; it's probably over 50VAC, though. A good wiring diagram is your friend, now.

This should help: http://mototh.com/files/kawasaki/ER6n/Kawasaki-ER6n-Service-Manual-EN.pdf
 

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That does look unusual; usually the coils all discolor about the same all around. Find the connector that connects that to the rectifier/regulator - I'm guessing the other end of those three wires, left side of photo. Use an Ohmmeter to measure between each of them (1-2, 1-3, 2-3), and then measure them to the metal of the engine case. The first three readings should be between 0 and 2 (not many meters can measure that low, accurately), with all just about the same, and the reading to the case should be open, or nearly so. From the look of that, I suspect you will get something else.
 

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The reading is open when you touch nothing with the probes; the display varies on different meters. The outside of the case is probably clear coated, which would read as open; to get the same reading you got inside the case, you would have to pierce the coating with the probe tip. But, yeah, it seems the stator is shorted to the case, which is kind of what I expected from the discoloration.
 

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Is it normal for them to deteriorate like this? The bike is 9 years old and has over 40k on it, and as far as I know this is the original stator.
No, it is not normal, or shouldn't be. As you can see, the stator is just a lot of wire coiled around some iron posts - what can go wrong? Well, manufacturing defects. Burgman scooters had a rash of defective stators recently, but the one in my '70 CB450 is original. Check to see that any points where the wires are joined won't be close to touching anything else, and look closely for cracks in the coating on the wire; those are the two most common failures, and often occur during winding them.
 

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The input to the rectifier/regulator is a bridge rectifier, probably 3 phase. Three diodes that feed forward to the battery (+), and three that feed back from chassis ground. If you measure the three contacts of the connector with your meter in diode test mode, each to the other, it should read open, or nearly so. If you read each to ground with the positive wire (usually red) of the meter on the contact, it should still read open, or nearly so, but if you reverse the leads, it should read a diode (about 0.6) on every one. If you measure from each contact to the battery (+) terminal, red on the contact and black on the battery, you should read a diode, but if your reverse the leads, it should read open (or nearly so).
This shows a simple version: https://www.allaboutcircuits.com/textbook/semiconductors/chpt-3/rectifier-circuits/
Yours will have more components to support the regulator function, but those are the basics.
 

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I'd say your meter may be a bit off; fully charged, the battery should rest at about 12.6V, and your system seems to be charging it OK. Maybe compare to another meter?
 
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