New Markup.html

<!-- Change pfe-navigation-item to a div tag -->
<div slot="search" pfe-icon="web-search" id="utility-search">
  <!-- All of the "trigger" element and the two wrappers pfe-navigation-item and form can be deleted -->
      <form class="search-form" autocomplete="off" action="/en/search/node" method="post" id="redhat-www-solr-search-box" accept-charset="UTF-8">
        <div>
          <div class="container-inline form-wrapper" id="edit-basic">
            <div class="form-item form-type-textfield form-item-keys">
              <label class="element-invisible" for="edit-keys">Enter your keywords </label>
              <input maxlength="128" size="15" placeholder="Enter your search term" type="text" id="edit-keys" name="keys" value="" class="form-text" />
            </div>
            <div class="form-actions form-wrapper" id="edit-actions">
              <input type="submit" id="edit-submit" name="op" value="Search" class="form-submit" />
            </div>
          </div>
          <input type="hidden" name="form_build_id" value="form-GUQB-cYo5rkA9_PeXx5mvpqUZnW7ZQG7jGUDVkdsmz4" />
          <input type="hidden" name="form_id" value="redhat_www_solr_search_box" />
        </div>
      </form>
</div>

Old Markup.html

<pfe-navigation-item slot="search" pfe-icon="web-search" id="utility-search">
  <div slot="trigger">
    <a href="/en/search" data-analytics-region="utility" data-analytics="Search" data-analytics-label="Search">Search</a>
  </div>
  <div slot="tray" hidden>
    <div class="pfe-navigation-item__tray--container">
      <form class="search-form" autocomplete="off" action="/en/search/node" method="post" id="redhat-www-solr-search-box" accept-charset="UTF-8">
        <div>
          <div class="container-inline form-wrapper" id="edit-basic">
            <div class="form-item form-type-textfield form-item-keys">
              <label class="element-invisible" for="edit-keys">Enter your keywords </label>
              <input maxlength="128" size="15" placeholder="Enter your search term" type="text" id="edit-keys" name="keys" value="" class="form-text" />
            </div>
            <div class="form-actions form-wrapper" id="edit-actions">
              <input type="submit" id="edit-submit" name="op" value="Search" class="form-submit" />
            </div>
          </div>
          <input type="hidden" name="form_build_id" value="form-GUQB-cYo5rkA9_PeXx5mvpqUZnW7ZQG7jGUDVkdsmz4" />
          <input type="hidden" name="form_id" value="redhat_www_solr_search_box" />
        </div>
      </form>
    </div>
  </div>
</pfe-navigation-item>