Okay, wp's "Listview_Selected.zip" code works for me.
Then I tried again my project ..
As this is an old Delphi project of mine,
(written between 2000 and 2004 ..)
I converted it with the Lazarus conversion tools.
As there are several differences between LCL and VCL,
lots of code didn't work as intended.
The last days I edited a lot of code and .. however, this
ListView.ItemIndex:= i;
works now !
I only wonder why MY selection is BLUE
and the selection in wp's "Listview_Selected.zip" code is light GRAY ?