3.30. The lateral edge of a regular prism $A B C A_{1} B_{1} C_{1}$ is equal to $a$; the height of the base of the prism is also equal to $a$. Through point $A$, planes are drawn perpendicular to the lines $A B_{1}$ and $A C_{1}$, and through point $A_{1}$ - planes perpendicular to $A_{1} B$ and $A_{1} C$. Find the volume of the figure bounded by these four planes and the plane $B_{1} B C C_{1}$.
3.30. Let $M$ and $N$ be the midpoints of the edges $B_{1} C_{1}$ and $B C$. The planes of the considered spheres are symmetric relative to the plane $A A_{1} M N$. Take a point $K$ on the ray $M N$ such that $M K=2 M N$.

Since $A A_{1} M N$ is a square, $K A \perp A M$, and therefore, the line $A K$ is perpendicular to the plane $A B_{1} C_{1}$, i.e., $A K$ is the line of intersection of the considered planes passing through point $A$. Similarly, the line $A_{1} L$ is the line of intersection of the planes passing through point $A_{1}$. Since $B_{1} N$ is the projection of the line $A B_{1}$ onto the plane $B C C_{1}$, the plane passing through point $A$ and perpendicular to $A B_{1}$ intersects the plane $B C C_{1}$ along a line perpendicular to the line $B_{1} N$. Conducting similar reasoning for the other considered planes and considering that the triangles $B M C$ and $B_{1} N C_{1}$ are equilateral, we obtain that the considered planes cut out a rhombus on the plane $B C C_{1} B_{1}$, consisting of two equilateral triangles with side $K L = 3 a$. The area of this rhombus is $9 a^{2} \sqrt{3} / 2$. The desired figure is a quadrilateral pyramid, the base of which is this rhombus, and the vertex is the point $S$ of intersection of the lines $A K$ and $A_{1} L$. Since the distance from point $S$ to the line $K L$ is $3 a / 2$, the volume of this pyramid is $9 a^{3} \sqrt{3} / 4$.