Analytic equivalence of plane curve singularities yn +xαy +xβa(x) = 0

There are not many examples of complete analytical classification of specific families of singularities, even in the case of plane algebraic curves. In 1989, Kang and Kim published a paper on analytical classification of plane curve singularities yn+a(x)y+b(x) = 0, or, equivalently, yn+xαy+xβA(x) = 0 where A(x) is a unit in Ct{x}, α and β are integers, α _ n − 1 and β _ n. The classification was not complete in the most difficult case α n−1 = β n. In the present paper, the classification is extended also in this case, the proofs are improved and some gaps are removed.