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That is correct. And it was there but it somehow got disabled for mobile site. I had to control-clich to field to get rid of that mark for "disable for mobile site" and now it is visible again.
Still wonder how to turn off Search icon?
If you want to turn search completely off, just deactivate it in Toolbar Site menu.
I did that in ...path-to-site/studio/builder_menu.php?page=items&module=system&set=sys_toolbar_site but search icon still present....
Try to clear cache.
