OK, I have rechecked my configurations. It appears, that network-manager is started, when I do a "systemctl enable NetworkManager". However, still /etc/ init.d/network-manager is missing. Dunno, if this is still needed.
I also discovered, that (maybe again?) the problem appears, that the applet , here exactly nm-applet got into a problem, when running in LXDE. This problem appeared in the past, but was fixed. Now it appears agian. When I am running plasma, then nm-applet gets access to the network-manager service, but when I start LXDE, I get the message, that the network-manger- service is not running. So it looks for me, like a rights problem. When I do start nm-applet as root in LXDE, then I get access to the network- manager-service, but when I do the same as a normal user, access is forbidden (something dbus and so). Please note, that I do not use a system with sudo, my installation is some years old, where sudo was not existent. However, this system is up-to-date debian/testing, and running well. I would not have changed to nemtwork-manager, because I was happy with wicd, but as wicd was removed from the repo, I chose the alternative. Thanks for looking at this bug. Best Hans