OK, after investigating wherever something is in the tables regarding Search i found some cadavers in the table modules. Here were three modules, one of them marked as deleted, referred as SearchResult-module.
I deleted them, i checked whether there is an entry in tabs, no....ok, lets start over.
Newly created page "Search Result", view by Registered Users, one Search Result Module on it, and.....nope.
When typing anything in the search box in the upper right corner (no extra SearchInput Module!) it does not show the Result page, When this page is shown, then the results are coming up and are correct.
If this wouldn't be enough...when placing a SEARCH INPUT Module on a page of your choice, it's working like desired.
But why the hack does this page not come up when searching over the SearchInput in the upper right corner defined probably by the skin? By the way, i'm using the standard Skin coming with DNN.....just in case....