As you may see, the search input field has the following code:

<input placeholder="Search..." onkeydown="return bx_search_on_type(event, 5, '#sys_search_form_quick', '#sys_search_results_quick', '#sys_search_form_quick', 1, 1, bx_site_search_complete);" onpaste="return bx_search_on_type(event, 5, '#sys_search_form_quick', '#sys_search_results_quick', '#sys_search_form_quick', 1, 1, bx_site_search_complete);" type="text" name="keyword" value="" data-frmt-24h="on" data-frmt-date="D MMM YYYY" data-frmt-datetime="D MMM YYYY h:mm:ss a" class="bx-def-font-inputs bx-form-input-text">

The 4th argument of the bx_search_on_type function is the # id of the area where the search result will be loaded. So you may change it in your variant, or locate #sys_search_form_quick div in the proper place on your page(s).