  For an interval $E=[a,b]$ on the real line, let $\mu$ be either the
equilibrium measure, or the normalized Lebesgue measure of $E$, and let
$V^{\mu}$ denote the associated logarithmic potential. In the present paper, we
construct a function $f$ which is analytic on $E$ and possesses four branch
points of second order outside of $E$ such that the family of the admissible
compacta of $f$ has no minimizing elements with regard to the extremal
theoretic-potential problem, in the external field equals $V^{-\mu}$.
