'81 GS 550, 12.7 volts @ battery with engine running???

[SuperCell]

This is not enough, it should probably be around 14. There is absolutely no change in voltage when the engine speed is increased. This probably explains why my headlight nearly died last night, and I had a dead battery this morning.

What parts do I need to replace to fix this problem?

[ddaum]

Sounds like you got the electrical bug. Your bike's symptoms could be the result of a lot of things that you need to narrow down to one. First, go through your entire electrical system inspecting/cleaning/preserving all electrical grounds and connections with contact cleaner and dielectric grease. You will be happy you did this in the future.

Once that is done, if the problem persists, go out and buy a quality electrical multimeter and read the instructions on how to properly use it.

Then read/print the STATOR PAPERS. I printed them and put them in a binder for future use. The stator papers at first glance may look intimidating, but it's really easy to follow and successfully diagnose your problem if you just read, go slowly, and ask questions if you need to. Always WRITE down your results from electrical testing so you can use them to ask questions and refer to later.

Don't run out ad buy a new stator or R/R right away, find the problem. It could save you a good chunk of cash.

Good luck.

[SuperCell]

Thanks a lot.

The funny thing is, the battery does seem to hold it's charge for a decent amount of time. I could probably just throw it on charge a couple hours every few days and get away with it for now.

[ddaum]

Thanks a lot.

The funny thing is, the battery does seem to hold it's charge for a decent amount of time. I could probably just throw it on charge a couple hours every few days and get away with it for now.

This is true but if you do that, you are running the risk of damaging more components. This is taken directly from the stator papers FAQ:
The GS Resources' 1st Law on Faulty Electrics:

Most any electrical problem starts with defective wiring or dirty connections. As a result of this, one by one the more expensive components will fail. Only then you will notice that something's wrong.

If you have a car or other transportation, I'd park the bike and troubleshoot/fix the problem before it gets too expensive.

[83'GK]

Check ALL the connections, and especially the grounds....