  A famous result of Stieltjes relates the zeroes of the classical orthogonal
polynomials with the configurations of points on the line that minimize a
suitable energy. The energy has logarithmic interactions and an external field
whose exponential is related to the weight of the classical orthogonal
polynomials.
  The optimal configuration satisfies an algebraic set of equations: we call
this set of algebraic equations the Stieltjes--Fekete problem or equivalently
the Stieltjes--Bethe equations. In this work we consider the Stieltjes-Fekete
problem when the derivative of the external field is an arbitrary rational
complex function. We show that its solutions are in one-to-one correspondence
with the zeroes of certain non-hermitean orthogonal polynomials that satisfy an
excess of orthogonality conditions and are thus termed "degenerate". This
generalizes the original result of Stieltjes.
