Thanks for that nicely written reply.
I did the 'root find grep' search on "gnome-shutdown.svg", the name of the icon in use now, and got only one line showing the root path to the file:
Now oddly enough in usr/share/applications there resides a file titled "System Shutdown Dialog" which is a desktop configuration file. The very same icon is associated with it. If you double click the icon the shutdown dialog box appears offering the ususal choices of Suspend, Hibernate, Restart, Cancel, and Shutdown. However this shutdown dialog box is shaped differently that the box which appears if you select the shutdown icon on the panel.
I changed the icon under Properties of this System Shutdown Dialog file yesterday. The icon changed as I desired at the file in the folder but the desktop panel icon showed no change. Used root, re-booted, etc. So I changed it back to what it was originally, gnome-shutdown.svg.
All my searching yesterday pointed to this file as the one and only System Shutdown Dialog. Strange that engaging it in it's folder brings up a different dialog box than when it is engaged on the desktop panel. Also strange that changing the icon in it's properties seems to have no affect outside the folder it resides in.
And I'm thinking that grep should have showed the icon associated with the system shutdown dialog. Yes?