  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 $r > 0$.
