The main aim of this paper is the study of the general solution of the exceptional Hermite differential equation with fixed partition \(\lambda = (1)\) and the construction of minimal surfaces associated with this solution. We derive a linear second-order ordinary differential equation associated with a specific family of exceptional polynomials of codimension two. We show that these polynomials can be expressed in terms of classical Hermite polynomials. Based on this fact, we demonstrate that there exists a link between the norm of an exceptional Hermite polynomial and the gap sequence arising from the partition used to construct this polynomial. We find the general analytic solution of the exceptional Hermite differential equation which has no gap in its spectrum. We show that the spectrum is complemented by non-polynomial solutions. We present an implementation of the obtained results for the surfaces expressed in terms of the general solution making use of the classical Enneper–Weierstrass formula for the immersion in the Euclidean space \({\mathbb {E}}^3\), leading to minimal surfaces. Three-dimensional displays of these surfaces are presented.

本文的主要目的是研究具有固定分区\（\ lambda =（1）\）的例外Hermite微分方程的一般解以及与此解决方案相关的最小表面的构造。我们推导了一个线性二阶常微分方程，它与一维二阶例外多项式的特定族相关。我们证明了这些多项式可以用经典Hermite多项式表示。基于这一事实，我们证明例外Hermite多项式的范数与用于构造该多项式的分区所产生的间隙序列之间存在联系。我们找到了例外的Hermite微分方程的一般解析解，该方程在频谱上没有间隙。我们表明，频谱是由非多项式解决方案补充的。\（{\ mathbb {E}} ^ 3 \），从而导致最小的表面。这些表面的三维显示。