I can find many modification of the JungAbhyankar theorem. I can even find a new proof of the theorem (by K. Kiyek and J. L. Vicente). But I cannot find the original statement. Does any one know which paper/book is it in? Where can I access it?
The original papers are accessible online:
Jung's paper is devoted exactly to this result, whereas Abhyankar's gives a more contextualized explanation (his was an unsuccessful attempt to pass to characteristic $p$); I think the version of AbhyankarJung in Abhyankar is Theorem 3 (but it might be worth studying the paper carefully). In Jung's statement, which I reproduce here, the field $K$ is defined as $K=\mathbb{C}(x,y)[z]/(f)$ for some irreducible polynomial $f$ (which is implicitly assumed to involve all three variables).
My translation:
This seems to be equivalent, in the formulation usual in more recent papers, to the following (I use $\mathbb{C}\{x\}$ to denote convergent power series):
Abhyankar considers the case of $n$ variables over an algebraically closed field of characteristic zero. 


The fractional power series solutions remain convergent in several variables. In fact, if you replace the ring of convergent power series by any Henselian ring $H$ of power series series satisfying some stability properties, then the fractional power series solution remain in the ring. See for instance http://arxiv.org/abs/1103.2559 

