I'm trying to customise the mark-up of the default search results page in Drupal 6. Specifically I'd like to remove the search box and the title from the page - I know I can hide it with CSS, but I'd rather it wasn't rendered in the first place.

Ideally, in the same why that you theme a particular content type node by copying the node.tpl.php and renaming it to something like node-blog.tpl.php and then amending the mark-up accordingly - is there an equivalent way to do this for the search results page?