A classification of generalized second-order Bottcher's equations

The authors of the article summarize their research on the solution of generalized Böttcher's equation of the second order from two arguments. The aim of the study is to describe a class of smooth solutions of these equations defined on a certain conic domain with a vertex at the origin. The method of direct description of the orbits of the action of a general linear group on the space of tensors of type (2,1), which are symmetric with respect to covariant indices, proved to be resolving. In the article, structural theorems on the structure of orbits were proved (Theorems 1-4). It was also proved that any generalized second-order Böttcher's equation belongs to one of the thirteen types of equations corresponding to tensors, which were called canonical by the authors (Theorem 5). In this article, part of the generalized Böttcher's equation is solved completely and the rest is reduced to four one-parameter and two two-parameter families of the functional Schröder's equations from one variable. The study also presents partial solutions to these equations.