2 system folders

I find that I have two System Folders: The normal one in Macintosh HD and a second one in a Previous System on my Desktop. I've been try to delete the second one but fail with alerts that I don't have enough permissions or that the system file is required in OSX and cannot be deleted. I've have tried all of the Finder based delete options (Secure Delete, option-delete, option-shift-delete, and all of the standard Terminal commands to force empty Trash. None work. There are two folders in the Previous System file (private and System). 'private' seems to be in use. Many help in sorting this out would be appreciated.