I see, I was looking for the error you reported, not paying attention to the results.
Sorry, the code for the next link:
<a href="<?php echo $search_rs->getNextPageLink(); ?><?php echo(isset($_POST['search'])?"&". $_POST['search']:""); ?>">Next</a>
should be:
<a href="<?php echo $search_rs->getNextPageLink(); ?><?php echo(isset($_POST['search'])?"&search=". $_POST['search']:""); ?>">Next</a>