MZ/MuZ Riders

problems like this can be caused by bad solder also

It definitely shouldnt be rigidly mounted. I replaced mine with an electronic one years ago, but as far as I remember it was fixed on the right side with a rubber "shock absorber" similar to that at the bottom of the rectifier. At the left side it was in a foam rubber sleeve.
Edit: It is part no. 30-34.037 Fastening rubber for regulator cut-out.

Yes that part rotted away on my ETZ250. I squirted a bunch of silicone around that end an let it set. So far, so good. Regulator still works and keeps the original 1985 battery charged.

Yes the Battery life has amazed me. I have changed quite a few batteries in other bikes over the same period. The battery supplied with the ETZ when I bought it was a Yuasa standard lead acid battery. Twenty plus years ago I came out to ride it to work but the battery was flat. So I pushed it into the back of the shed. There it stayed until a few years ago. When I pulled it out to recommission it I noticed that the headlight and ignition had been left on all that time. The battery was about quarter full. I topped it up with tap water and put it on the charger. It held charge, the bike started and has been working fine ever since. I know it does not make sense.