This paper deals with the blow-up problem for a model semilinear equation of the type \[ \text{div}(A(z)\nabla u)= e^u \] in a simply connected domain \(\Omega\subset \mathbb{C}^1\). \(A(z)\) is a \(2\times 2\) symmetric uniformly elliptic matrix with measurable entries and \(\text{det\,}A=1\). The authors show that the Liouville-Bieberbach function solves the problem under consideration in the case when \(A\) is generated by a special Beltrami coefficient.