The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system

The distribution of the zeros of the Hermite-Padé polynomials of the first kind for a pair of functions with an arbitrary even number of common branch points lying on the real axis is investigated under the assumption that this pair of functions forms a generalized complex Nikishin system. It is proved (Theorem 1) that the zeros have a limiting distribution, which coincides with the equilibrium measure of a certain compact set having the $\mathscr S$-property in a harmonic external field. The existence problem for $\mathscr S$-compact sets is solved in Theorem 2.
The main idea of the proof of Theorem 1 consists in replacing a vector equilibrium problem in potential theory by a scalar problem with an external field and then using the general Gonchar-Rakhmanov method, which was worked out in the solution of the `$1/9$'-conjecture.
The relation of the result obtained here to some results and conjectures due to Nuttall is discussed.
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