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Hi, The iPod is just working with power cable - PC or charger. After unpowering and reconnecting it is booting new. I know how to get into Diag Mode. How can i see if ich have a problem with the battery or the charging electronic?

More then likely your battery has lost the ability to hold a charge. The fact that it works while plugged in suggest that the charge port, and the charging circuit are most lilely ok. Battery replacements are commonly needed for iPod 6gen if it was never replaced by now. Note that if you chose to replace it yourself, that battery is soldered to the motherboard I hope this help