Module assignment to single items but not to the category

Hi! I am in trouble with some modules display. I would like to get this:

1. K2 latest items (2 columns) with no modules displayed like this:

http://demo.gavick.com/joomla25/startup ... wo-columns

and at the same time

2. Single item (related to k2 latest items) with the modules displayed like this

http://demo.gavick.com/joomla25/startup ... ign-agency

You do this in the demo. How can I get it? The problem is that if I assign the modules to the "latest news" menu item (Type: latest items from (one or more) users or categories) the modules are shown also in point 1.

I hope my explanation is clear! Thank you for your help!

Hi,
easiest way to achieve same look as demo is install somewhere else our quicstart package and then compare settings.