Solutions of some systems of functional equations related to complex, double, and dual numbers

In this paper, we solve the problem on the embedding of three two-metric, phenomenologically symmetric geometries of two sets of rank $(3,2)$ related to complex, double, and dual numbers, into a two-metric, phenomenologically symmetric geometry of two sets of rank $(4,2)$ determined by a functions of two points $f=(x\xi +y\mu + \rho, x\eta +y\nu + \tau)$. The problem is reduced to the search for nondegeenerate solutions of three special systems of functional equations immediately related to complex, double, and dual numbers.