  This paper extends approach of recent author's paper devoted to special
classes of exact solutions of the static Maxwell system in inhomogeneous
isotropic media and new generalizations of the Cauchy-Riemann system in
$\mathbb R^3$. Two families of generalizations of the Cauchy-Riemann system
with variable coefficients in $\mathbb R^4$ are presented in the context of
non-Euclidean geometry. Analytic models of a wide range of potential meridional
vector fields in $\mathbb R^4$ are characterized using a family of Vekua type
systems in cylindrical coordinates. The specifics of meridional fields allows
us to introduce the concept of four-dimensional $\alpha$-meridional mappings of
the first and second kind depending on the values of a real parameter $\alpha$.
In case $\alpha=2$ tools of the radially holomorphic potential in $\mathbb R^4$
are developed in the context of generalized axially symmetric potential theory
(GASPT). Analytic models of potential meridional fields in $\mathbb R^4$
generated by Bessel functions of the first kind of integer order and
quaternionic argument are described in case $\alpha=2$. In case $\alpha =0$ the
geometric specifics of four-dimensional harmonic meridional mappings of the
second kind is demonstrated explicitly in the context of the theory of gradient
dynamical systems with harmonic potential.
