Is there a way that our website add a textbox then integrate it to google ?
Of course the listing search all is below from our own sites, and to change or add our logo to google listing too.
I think you can do that with adsense. You will even get some money if your visitors click the google ads!
wow there's many search google.
Google Free WebSearch
Google AdSense for search
Public Service Search
Google Search Appliance
which one should i take, i want google have my logo on the search engine also and only search at my sites
To use your logo in the search page you need to have an Adsense account. If you want Google to search on your site you can use 'site:yoursite.com search phrase here'