if you look closely at the URL for a search that is working correctly, notice that it contains the following URL parameters:
Search.x=5
Search.y=7
Search=Search
the trigger:
Trigger: isset($_GET["Search_x"]) && $_GET["Search_x"]
is checking for the Search.x parameter, this is the x coordinates of where the image button is pressed.
your links that are not working:
product_group.php?S_ProductCategoryID=1&WADbSearch 1=1
dont work because the Search.x parameter is missing.
change that to include the Search.x parameter:
product_group.php?S_ProductCategoryID=1&Search.x=1