Sidebar for search results

Is there a way to display a special “widget area” (aka sidebar) just for the search result page?

The reason:
I use the “Displayed Everywhere” widget area to display a search form.

And therefor I have now twice a search form at the search results page:
one in the sidebar and one on top of the search results.

Thanks, Ismael, I would like to have a sidebar only for the search results page, where I can put widgets in it, that appear nowhere else.
So “Display Everywhere” is not the right way for me.

Thanks, Ismael, but I don’t understand what you mean with “new template”.

Can I create a new “widget area” for the search results page and its widgets and … then what do I need to do next please?

For example I created a new “widget area”, called “area_searchsidebar”, but when I add in search.php as you suggested

get_sidebar('area_searchsidebar');

… nothing happens, my custom sidebar is not shown.

thanks for your help, Ismael. I am not familiar with templates, never have understood “Widget Logic” plugin, but I found a similar one with more “visual” and for me easier to understand settings – “Widget Options“, and with that I got it easily to work by creating my desired widgets in the PAGE area and setting them via “Widget Options” to “Search”.