Hello,
I would like to let you know, it is possible to convert browser based Flash/Flex application to Air application. However, that Air application is an installable file not a browser. You can have a webview within Air application but building a browser that can run Air file is virtually impossible. There are couple of more options you have which might not be immediate. Html5 and Apache Royale for converting browser based application to flash independent browser application. Please PM me for more info.
Thanks