The Böttcher functional equation and one of its real generalizations are considered. It is shown that, in some situations, after finding a solution of the generalized equation, other solutions can also be obtained. For example, a three-parameter family of real functional equations for a function of two arguments is described, for which solutions are found. The generalization described has wide application. Many quantities after an appropriately introduced parameterization satisfy the generalized Böttcher equation as functions of parameters. As an illustration, two-parametric families generated by the determinant of a linear combination of second-order matrices are presented. It is shown that the parameterized Poisson integral as a function of its parameters satisfies the generalized Böttcher equation. This made it possible to calculate the Poisson integral and the Euler integral in a new way. In addition, the calculation of the Poisson integral by the method of integral sums is described.