The problem is solved now.
Here is how the theme installing company that solved the problem described the solution:
"Hi,
It is miss two file at controllers\admin\ I upload it to your hosting, and now site is working good again.
Best regards."
I wonder why did these files not add up when I copied the controllers folder from prestashop 1.5.6.0 to the controllers folder myself in ftp (I used filezilla for this.).
I admire a lot the amount of time and attention given to this problem, especially by Vekia.
Thanks.