  We calculate the Faddeev-Popov operator corresponding to the maximally
Abelian gauge for gauge group SU(N). Specializing to SU(2) we look for explicit
zero modes of this operator. Within an illuminating toy model (Yang-Mills
mechanics) the problem can be completely solved and understood. In the field
theory case we are able to find an analytic expression for a normalizable zero
mode in the background of a single `t Hooft instanton. Accordingly, such an
instanton corresponds to a horizon configuration in the maximally Abelian
gauge. Possible physical implications are discussed.
