icedowl - 9:20 am on Sep 19, 2004 (gmt 0)

From Google AdSense FAQ: 8. Can I display AdSense for search and Google ads on the same web page? Yes. You can place an AdSense for search box on any page that meets our program policies, even if you're already showing Google ads on that page. Searches from the AdSense for search box will lead to a Google-hosted results page that is also monetized with Google ads, allowing you to monetize all aspects of your site. This doesn't allow the display of Ads from Content and Websearch are on the same page. It allows you to place a "search box" on a page that has Google ads.

Yes it does allow both on the same page! Read both the "FAQ" page and the "what's new" page. Google changed the name of 'Websearch':

New name for WebSearch

We're dropping the 'WebSearch' from 'WebSearch plus AdSense for search.' This monetized search feature will now be referred to simply as AdSense for search - but all of the same great features remain. For more information on providing customized Google internet and site search to your users and earning money from clicks on targeted ads, please read our FAQ.

Besides, I think invisible was referring to a different sort of search engine. One that is not on Google's servers.