  Problems, related to the determination of the minimal number of balls that
generate a shadow at a fixed point in the multi-dimensional Euclidean space $
\mathbb{R}^n $, are considered in present work. Here, the statement "a system
of balls generate shadow at a point" means that any line passing through the
point intersects at least one ball of the system. New properties of
pairwise-disjoint balls that do not contain a fixed point inside of a sphere in
space $\mathbb{R}^3$, centered on the sphere, and generate shadow at the point
are established. And a system of $n+1$ pairwise-disjoint balls with equal radii
in $\mathbb{R}^n$, $ n\ge 3$, that do not contain a fixed point of the space
and generate shadow at the point is constructed in the work.
