Officially, you cannot have an anonymous Facebook page profile, but off the record, you can. 16% of Facebook users (accounts) are duplicates or fake. To make it difficult for other people to track down your "anonymous" account, try the following:
Configure your profile to be invisible to the general public (except friends and friends of friends) from the "Facebook Privacy Settings and Tools" menu;
You can hide your profile (make it completely invisible) from specific users by blocking them;
Deactivate your account;
Use the "Privacy Preferences" menu to hide the majority of your profile info.
Whatever you do, your name and profile picture will always be visible on your personal profile. The problem with creating a secret Facebook account is that you're breaking the Facebook rules and your account might be disabled. Facebooks community' policy requires full identity disclosure, specifically the real first and last name (no initials). Using Facebook with a fake name could result in account suspension.