Two questions actually:
1) Does there exist a 128-bit number whose md5 hash is itself?
X == md5(X) - does 'X' exist and can it be found without brute force?
2) Does there exist two 128-bit md5 digests that hash to each other?
Y == md5(X) && X == md5(Y) - do 'X' and 'Y' exist and can they be found without brute force?
This is not the same as the Kember Identity Search.
Consider the differences of the following cases:
md5(X) == X
For this to be true, X must be a 128-bit value.
This is not the same as the following:
bin2hex(md5('string')) == 'string'
Which is what the Kember Identity Search is actually seeking. If you take a look at any of the search implementations on their site, you can easily see that they are working with 32-character strings, not with 128-bit numbers, as the input to the md5 function, and thus are not seeking md5(X) == X.
I am not the first to point this out either, you might find This Article Directly Targeting The "Kember Identity" by Kris Thompson enlightening.