More than one search results page template

You could make a page template http://codex.wordpress.org/Pages#Creating_Your_Own_Page_Templates

And then you could post some data via $_GET /search2/?search=xxx to that page and do a custom wp_query where you use ‘s=” . $_GET[“search’]
http://codex.wordpress.org/Class_Reference/WP_Query

Something like this:

$args = array(
   's' => $_GET['search']
);
$the_query = new WP_Query( $args );

// The Loop
while ( $the_query->have_posts() ) : $the_query->the_post();
    echo '<li>';
    the_title();
    echo '</li>';
endwhile;