A characterization of isometries of CAT(0)-space as maps preserving diagonal tube

We give positive answers for questions by Berestovskii. Namely, we prove that every bijection of locally compact geodesically complete and connected at infinity CAT(0)-space $X$ onto itself preserving some fixed distance or satellite relations is an isometry of this space. The proof of this theorem is based on another result stated by Berestovskii as a problem: the metric of the space $X$ may be recovered from its diagonal tube corresponding to an arbitrary number $r > 0$.