I used to have a Facebook personal user profile, with which, among other things, I managed a Page for an organization. Later, I decided I didn't want to maintain a personal presence on Facebook - though I still needed to administer the Page. So I created a new anonymous fake user profile with the same name of the organization, and added that as administrator of the Page. I then deleted (rather than deactivated) my real user profile. Facebook's deletion service takes 2 weeks before they finally delete the profile.
However, creating a personal user profile with anything other than one's own name violates Facebook's terms of service. That carries the danger that they could one day close the user profile. The web is replete with stories of them actually doing so. So I began to seek a better solution.
Facebook has another kind of account, known as a "business account" which is intended strictly for those who want to create a Facebook Page or Ad, but don't want to maintain a personal user profile. It is not permissible to have both a personal user profile and a business account, and it is not permissible to convert a personal user profile back to being a business account. All of this is explained at http://www.facebook.com/help/?search=business%20account.
What they do not explain is how to create a new business account and let that administer an existing Page. I learned this by performing an experiment. Opening Facebook in a Chrome Incognito session, I followed Facebook's instructions for creating a new Page, giving this new Page a nonsensical name. Then, following the same instructions, I created the business account to administer the nonsensical Page, using an email address that was not in Facebook's records. Then, in another browser, I edited the organizational Page with the fake personal user profile that administered it, and added the newly created "business account" email address as an additional administrator of the Page. After checking that this "business account" could actually administer the Page, I deleted the fictional Page that I needed in order to create the "business account", then deactivated (not deleted, for now) the fake personal user profile I had used to administer the real organizational Page.
To sum things up, I now have a "business account" that can administer an organizational Page, and am in full abidance with Facebook's terms of service. Or will be, when I delete the deactivated fictitious personal user profile. For safety's sake, I'm waiting a bit before doing that.
It's just a pity that in order to abide by Facebook's rules you need to break them.