Let us consider properties of $C^1$-solutions of a special class of axially symmetric generalizations of the Cauchy-Riemann system with variable coefficients in $\mathbb R^3$ $$ \left\{ \begin{array}{l} (x_1^2+x_2^2)(\frac{\partial{u_0}}{\partial{x_0}}- \frac{\partial{u_1}}{\partial{x_1}}-\frac{\partial{u_2}}{\partial{x_2}})+ \alpha(x_1u_1+x_2u_2) = 0, \\ \frac{\partial{u_0}}{\partial{x_1}}=-\frac{\partial{u_1}}{\partial{x_0}}, \ \ \ \frac{\partial{u_0}}{\partial{x_2}}=-\frac{\partial{u_2}}{\partial{x_0}}, \\ \frac{\partial{u_1}}{\partial{x_2}}=\ \ \frac{\partial{u_2}}{\partial{x_1}}. \end{array} \right. $$ Assume that $\alpha \in \mathbf{R}$. The scalar potential $ h= h(x_0,x_1,x_2)$ of every irrotational vector field $\vec V \equiv (u_0, -u_1, -u_2)$ in simply connected open domains $\Lambda \subset \mathbf{R}^3$, where $ \vec V(x_0,x_1,x_2) \equiv \mathrm{grad} \ h(x_0,x_1,x_2)$, satisfies the following second order elliptic equation: $$ (x_1^2+x_2^2)\Delta{h} - \alpha \left( x_1\frac{\partial{h}}{\partial{x_1}} + x_2\frac{\partial{h}}{\partial{x_2}}\right) =0. $$ Meanwhile, transformation of Cartesian coordinates $(x_0,x_1,x_2)$ into cylindrical coordinates $(x_0, \theta, \rho)$, where $\rho \equiv \sqrt{x_1^2+x_2^2}$, $\theta\equiv\arccos \frac{x_1}{\rho}$, under conditions $\rho >0$, $ u_{\rho} = - \frac{\partial{h}}{\partial{\rho}}$, $ u_1 = \frac{x_1}{\rho} u_{\rho}$, $u_2 = \frac{x_2}{\rho} u_{\rho}$ leads to $C^1$-solutions of a special class of Vekua-type systems: $$ \left\{ \begin{array}{l} \rho ( \frac{\partial{u_0}}{\partial{x_0}} - \frac{\partial{u_{\rho}}}{\partial{\rho}} ) + (\alpha -1) u_{\rho} = 0, \\ \frac{\partial{u_0}}{\partial{\rho}}=-\frac{\partial{u_{\rho}}}{\partial{x_0}}. \end{array} \right. $$ Fueter's construction is provided in case $\alpha=1$. As seen, the set of $C^1$-solutions in general case $\alpha \in \mathbf{R}$ is reduced to the set of $C^2$-solutions of the elliptic Euler-Poisson-Darboux equation: $$ \rho \left( \frac{\partial{^2}{h}}{\partial{x_0}^2} + \frac{\partial {^2}{h}}{\partial{\rho}^2} \right) - (\alpha -1) \frac{\partial{h}}{\partial{\rho}} = 0. $$ Let $I_H, II_H, III_H$ be principal invariants of the Hessian matrix $\mathbf{H}(h)$ with components $ \mathbf{H}_{l m}(h) = \frac{\partial^2{h}}{\partial{x_l} \partial{x_m}} $ $ (l, m = 0,1,2)$, where the scalar potential $h$ satisfies the elliptic Euler-Poisson-Darboux equation. Theorem. Roots of the characteristic equation $\lambda^3 - I_H \lambda^2 + II_H \lambda - III_H = 0$ are explicitly given by formulas: $ \quad \quad \lambda_{0} = -\frac{u_{\rho}}{\rho}, \quad \quad \lambda_{1, 2} = -\frac{(\alpha -1)}{2} \frac{ u_{\rho}}{ \rho} \pm$ $ \sqrt{ \frac{(\alpha -1)^2}{4} \left( \frac{u_{\rho}}{ \rho} \right)^2 - (\alpha -1) \frac{ u_{\rho}}{\rho} \frac{\partial{u_{\rho}}}{\partial{\rho}} + \left( \frac{\partial{u_{\rho}}}{\partial{x_0}}\right)^2 + \left( \frac{\partial{u_{\rho}}}{\partial{\rho}} \right)^2}.$ The main geometric properties of a special class of axially symmetric continuously differentiable mappings $\vec V \equiv (u_0, -u_1, -u_2): \Lambda \rightarrow \mathbb{R}^3$ are presented.