  Given a birational normal extension S of a two-dimensional local regular ring
R, we describe all the equisingularity types of the complete ideals J in R
whose blowing-up has some point at which the local ring is analytically
isomorphic to S.
  The problem of classifying the germs of such normal surface singularities was
already posed by Spivakovsky (Ann. of Math. 1990). This problem has two parts:
discrete and continous. The continous part is to some extent equivalent to the
problem of the moduli of plane curve singularities, while the main result of
this paper solves completely the discrete part.
