I found an article that can help in my investigations.
The problem is solved for the following system of vectors, which is shown in the figure.
The methodology described in the article allows, using simple numerical calculations on a computer, to determine the angles $\psi$, $\psi$'and $\phi$, $\phi$'.
All other angles ($\alpha_2$,$\alpha_3$,$\alpha_4$,$\alpha_5$,$\mu$) are known.
The coordinates of points $B_1$,$B_2$,$B_3$ and $C_1$,$C_2$,$C_3$ are also known.
I do not know how, knowing these parameters, it is possible to determine the coordinates of points $A_1$,$A_2$,$A_3$.
In the same article on page 525 there is such a sentence:
But I do not understand how to do this. I ask for help.