I have tried out a variety of scenarios to solve the problem: Move the adapter to other outlets in the home, connecting to another LAN port of the Fritz box, changing the energy management in the Fritz box for the LAN port of Gigabit! - to Fast Ethernet speed, replacing the LAN cable etc. Nothing has helped.
Finally, I could solve the problem by, that I have a LAN switch interposed with its own power supply. The LAN cable Powerline adapter at the Fritz! Box thus lies not directly in the LAN port of the Fritz! Box, but in a LAN port of the switch, which in turn is connected to the LAN port of the Fritz! Box. Since then, the connection is stable, the LAN LED on the Powerline adapter is lit continuously.
Apparently there is any problem with the power supply of the powerline adapters in combination with the Fritz! Box 6490th