From the text: S. S. Goncharov and many of his students have found descriptions for autostable elements in many natural classes of structures (Boolean algebras, linear orderings, abelian \(p\)-groups etc.). The main tools in those investigations were some generalized constructions for proving nonautostability, in particular a rather general and non-trivial branching theorem of Goncharov.
In this article, we suggest a new version of this theorem. This version turns out to be applicable to new classes of objects, which were not studied earlier. Using this version, we then describe autostable Boolean algebras with distinguished ideals. This version has also been used to describe autostable trees.