The main problem considered in the paper is the following: find \(f(z,\overline{z})\) with compact support in \({\mathbb C}\) and \(0\leq f\leq L\) almost everywhere, given \(a_{mn}=\int_{\mathbb C} z^m\overline{z}^nf(z,\overline{z})d A(z)\) for \(m,n\in{\mathbb N}\), where \(dA\) is the planar Lebesgue measure. The set of solutions is convex and is therefore characterized by the extremal solutions. These solutions are characterized by finitely many moments and are therefore also called degenerate solutions. This paper is devoted to describe the set of all degenerate solutions. These solutions turn out to be of the form \(L\) times the indicator function \(\chi_\Omega\) of some specific bounded open set in \(\Omega\subset{\mathbb C}\) with a real analytic boundary. The proof relies on the use of the principal function of a hyponormal operator whose self-commutor has rank one (i.e., an operators \(T\) for which the self-commutor \([T^*,T]=\xi\otimes\xi\geq 0\) with \(\xi\) a nontrivial vector), and on the kernel function for \(\Omega\): \[ E_\Omega(z,w)= \exp\left[{{-1}\over{\pi}} \int_\Omega{{dA(\zeta)}\over{(\overline{\zeta}-\overline{z})(\zeta-w)}}\right]. \] The interplay between complex analysis, operator theory, and moment problems leads to alternative characterizations of the degenerate solutions.