in the configuration my homepage is neesconsult.de.
But what should I change in the template to get the right behavior?

Yes the home for german is the main domain, and the home of the english at my site is it /en.

But what can I change? I do not want to have two domains,

Almost everything functions on my page www.neesconsult.de/ .
If i select english, the english page is shown.
The language is kept, when I select another menu or content. Only when I select the "home"-page, the german page comes back. This occures, when I change the language on a page different from the home-page.

What can I do?