This paper extends approach of our recent paper together with Kähler to building special classes of exact solutions of the static Maxwell system in inhomogeneous isotropic media by means of different generalizations of the Cauchy–Riemann system with variable coefficients. A new class of three-dimensional solutions of the static Maxwell system in some special cylindrically layered media is obtained using class of exact solutions of the elliptic Euler–Poisson–Darboux equation in cylindrical coordinates. The principal invariants of the electric field gradient tensor within a wide range of meridional fields are described using a family of Vekua type systems in cylindrical coordinates. Analytic models of meridional electrostatic fields in accordance with different generalizations of the Cauchy–Riemann system with variable coefficients allow us to introduce the concept of \(\alpha \)-meridional mappings of the first and second kind depending on the values of a real parameter \(\alpha \). In particular, in case \(\alpha =0\), geometric properties of harmonic meridional mappings of the second kind are demonstrated explicitly within meridional fields in homogeneous media.

本文扩展了我们最近与 Kähler 一起发表的论文的方法，通过具有可变系数的 Cauchy-Riemann 系统的不同推广来构建非均匀各向同性介质中静态 Maxwell 系统的特殊类型的精确解。利用圆柱坐标系中椭圆欧拉-泊松-达布方程的精确解，得到了一类新的静态麦克斯韦系统在一些特殊圆柱层状介质中的三维解。使用圆柱坐标系中的 Vekua 类型系统族描述了在大范围子午场内的电场梯度张量的主要不变量。\(\alpha \) -第一类和第二类经向映射取决于实参数\(\alpha \) 的值。特别是，在\(\alpha =0\) 的情况下，第二类调和子午映射的几何特性在均匀介质的子午场中得到了明确的证明。