You can declare your Power or Income to be up to 1 higher than it actually is and you can always elect to make it lower. But you can't use any special Illuminati powers.
Where's the fun in that?
The variant: In addition to the power and money changes listed above, before anybody draws any Illuminati to use, they draw an Illuminati and are able to use that Illuminati's special powers until their real Illuminati is exposed. If they draw Bavaria they can make a privileged attack for 5MB once per turn; if they draw Cthulhu they get +2 to destroy. If they draw the UFOs they can use their power twice *and* have the Secret Goal! However, as soon as the real Illuminati is exposed these special powers are gone and vanished.
Aside from this change, the rest of the rules for Hidden Illuminati apply.