Solution of three systems of functional equations related to complex, double and dual numbers

The article solves three special systems of functional equations arising in the problem of embedding of two-metric phenomenologically symmetric geometries of two sets of rank (3,2) associated with complex, double and dual numbers into a two-metric phenomenologically symmetric geometry of two sets of rank (4,2), which is affine group of transformations on the plane. We are looking for non-degenerate solutions of these systems, which in general are very difficult to find. The problem of determining the set of solutions to these systems, associated with a finite number of Jordan forms of second-order matrices, turned out to be much simpler and more meaningful in the mathematical sense. The solutions obtained have a direct connection with complex, double and dual numbers.