  Curves in the complex plane that satisfy the S-property were first introduced
by Stahl and they were further studied by Gonchar and Rakhmanov in the 1980s.
Rakhmanov recently showed the existence of curves with the S-property in a
harmonic external field by means of a max-min variational problem in
logarithmic potential theory. This is done in a fairly general setting, which
however does not include the important special case of an external field given
by the real part of a polynomial of degree greater than or equal to 2. In this
paper we give a detailed proof of the existence of a curve with the S-property
in the external field given by the real part of a polynomial V, within the
collection of all curves that connect two or more pre-assigned directions at
infinity in which the real part of V grows. Our method of proof is very much
based on the works of Rakhmanov on the max-min variational problem and of
Mart\'inez-Finkelshtein and Rakhmanov on critical measures.
