To remove the LINKS folder go to Start, Run, type regedit and navigate to registry key HKEY_CURRENT_USER\Software\Microsoft\Internet Explorer\Toolbar and Modify (or Create a new) REG_SZ value named "LinksFolderName" and leave the value blank.
Once you have done this, open IE and delete the links folder. This will prevent the links folder from being recreated (until IE and/or Favorites are updated) as Windows expects at least one letter in a folder name. It will put an entry as an IE Start and Search item which should not be removed via the registry or by running HijackThis or similar programs.
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