4. Let's highlight the complete squares: $\left\{\begin{array}{l}(x-2)^{2}+(y+1)^{2}=5, \\ (x-2)^{2}+(z-3)^{2}=13, \\ (y+1)^{2}+(z-3)^{2}=10 .\end{array}\right.$

Add all the equations $\left\{\begin{array}{c}(x-2)^{2}+(y+1)^{2}=5, \\ (x-2)^{2}+(z-3)^{2}=13, \\ (x-2)^{2}+(y+1)^{2}+(z-3)^{2}=14 .\end{array}\right.$

And, by subtracting the first and second from the third, we get:

$$
\left\{\begin{array}{l}
(x-2)^{2}=4 \\
(z-3)^{2}=9 \\
(y+1)^{2}=1
\end{array}\right.
$$

Solve the quadratic equations and write down all possible solutions. The system has eight solutions.
Answer: $(0 ; 0 ; 0),(0 ;-2 ; 0),(0 ; 0 ; 6),(0 ;-2 ; 6),(4 ; 0 ; 0),(4 ;-2 ; 0)$, 
$(4 ; 0 ; 6),(4 ;-2 ; 6)$.