Search field has no server side value until form is submited so button can't pull this value from form when it is rendered. You would need t do it either client side (js) or server side with event that fires after search form is submited (not sure whether there is one..)
actually the whole purpose was to add an anchor to the link of the SEARCH button. Since I didn't find a way to add this anchor I figured I had to use a free button but this time I can't pass variables.
SO I'm back to the first stage: how can I add an anchor to the result page with a search button please ?