On the solution of the Schroeder equation of a special form

The classification of the generalized second-order Böttcher's equations from two arguments, as a result of the classification theorem established earlier by the authors, was reduced to $13$ canonical functional equations corresponding to the orbits of the action of a general linear group on the space of tensors of type $(2,1)$ symmetric by covariant indices. The remaining five canonical equations were reduced to the real Schröder equations of one variable, which are interpreted as a question of the real conjugacy of the polynomial $t^2$ and some rational function (kernel). In this article, the triviality of any continuous solution is proved for four equations and the triviality of a real-analytical solution is proved for one remaining equation.