Scators, as defined by Manuel Fernández-Guasti and Felipe Zaldívar [

1], form a linear space with a specific multiplicative structure. In fact, we have two different structures: elliptic and hyperbolic. Namely, in the elliptic case, the scator product of scators

$\stackrel{o}{a}:=({a}_{0};{a}_{1},\dots ,{a}_{n})$ and

$\stackrel{o}{b}:=({b}_{0};{b}_{1},\dots ,{b}_{n})$ is given by

$\stackrel{o}{u}:=({u}_{0};{u}_{1},\dots ,{u}_{n})$, where

provided that

${a}_{0}\ne 0$ and

${b}_{0}\ne 0$ (more general case is presented and discussed in [

1]). In the hyperbolic case, the formula is very similar (all minuses are replaced by pluses). In principle, one can consider mixed cases as well. The scator product is non-distributive, although a distributive approach has been proposed recently [

2,

3]. The so-called restricted space (defined by

${a}_{0}^{2}>{a}_{k}^{2}$ for

$k=1,\phantom{\rule{4pt}{0ex}}\dots ,\phantom{\rule{4pt}{0ex}}n$) is an abelian group with respect to the scator product.

In the hyperbolic case, scators have potential physical applications related to generalizations of the special theory of relativity (breaking the Lorentz symmetry) [

4,

5]. The elliptic case is an interesting new example of (non-distributive) hypercomplex numbers [

6].

Any hypercomplex numbers, like quaternions or Clifford numbers, lead to a natural question of defining and finding anlogues of holomorphic functions. In this paper, following [

7], we focus on the most straightforward definition of holomorphicity, i.e., existence, at any point, of a direction-independent derivative. Fernández-Guasti derived a system of partial differential equations which assures scator differentiabiliy of this kind [

7]. They can be considered as a generalization of Cauchy–Riemann equations of standard complex analysis:

for all

$m\ne j$, where

m and

j take values from 1 to

n. Note that the last (nonlinear) equations appear only for

$n>1$. The generalized Cauchy–Riemann Equation (

2) consists of a set of linear equations (

n copies of the Cauchy–Riemann equations, in fact) and a set of nonlinear equations (for

$n>1$). The latter is the main difference with the standard case of complex holomorphic functions (i.e., the case

$n=1$).

In this paper, we are going to solve the open problem of finding all solutions of the system (

4) in the case

$n=2$. Until now only two particular solutions were reported: four-parameter family of linear affine functions [

7] and components exponential function [

8].