Local smooth conjugations of the Frobenius endomorphisms

In the paper, one of the generalizations of the Böttcher equation is considered. It turned out that the parametrized Poisson integral, as a function of its parameters, satisfies an equation of the type described. The structure theorem for splitting maps of the Frobenius endomorphism in a ring and in an algebra over it is proved. The real field case is considered. The generalized Böttcher equation is solved for classical two-dimensional algebras and for the Poisson algebra.