These buttons added by some third party plugin, disable all one by one and see which plugin adds these buttons, then remove it by disabling in Installed plugins section in the admin panel. 

Thank you Ross. Of course I did it. The only result is "YNO Social Publisher fault" (the buttons disappear if I disable this plugin) but, as I wrote, YNO technical assistance had reply that that bar don't belong to theyr plugin.

CSS Inspection made by myself doesn't seems to help. I  searched the code and the classes recalled by the code that are capable to change the position of the plugin seems don't belong to any of my site's site!

<div id="at4-share" class="at4-share addthis_32x32_style atss atss-left addthis-animated slideInLeft at4-show"> is the HTML

.at4-show....

.atss-left ...

are some of the classes that controls this box. 

The text search did no match on the entire site code! :(

Have you told them that after disabling their plugin the buttons dissappear? 

Yes, of course. No answer by now. Some days ago I upload a pair of free social plugins, lately removed. I think that one of them had made the mess. I'll update you with news (if any)