Solution of a system of functional equations associated with an affine group

Solution of the embedding problem for a two-metric phenomenologically symmetric geometry of rank (3,2) with the function g(x,y,ξ,η)=(g1,g2)=(xξ+y mu,xη+yν) into an affine two-metric phenomenologically symmetric geometry of rank (4,2) with the function f(x,y,ξ,η,μ,ν)=(f1,f2)=(xξ+yμ+ρ,xη+yν+τ) leads to the problem of establishing the existence of non-degenerate solutions to the corresponding system f(x¯,y¯,ξ¯,η¯,μ¯,ν¯)=χ(g(x,y,ξ,η),μ,ν) of two functional equations. This system is solved based on the fact that the functions g and f are previously known. This system is written explicitly as follows: x¯ξ¯+y¯μ¯+ρ¯=χ1(xξ+yμ,xη+yν,μ,ν), x¯η¯+y¯ν¯+τ¯=χ2(xξ+yμ,xη+yν,μ,ν). The main goal of this work is to find a general non-degenerate solution to this system. To solve the problem, we first differentiate with respect to the variables x, y and ξ, η, μ, ν, as a result we obtain a system of differential equations with a~matrix of coefficients A of the general form. It is proved that the matrix A can be reduced to Jordan form. Then a system of differential equations with such a Jordan matrix is solved. Returning to the original original system of functional equations, we find the additional restrictions. As a result, we arrive at a non-degenerate solution to the original system of functional equations.