KöMaL 2011 A538

In the 3-dimensional hyperbolic space there are given a plane $\mathcal{P}$ and four distinct lines $a_1$, $a_2$, $r_1$, $r_2$ in such positions that $a_1$ and $a_2$ are perpendicular to $\mathcal{P}$ , $r_1$ is coplanar with $a_1$, $r_2$ is coplanar with $a_2$, finally $r_1$ and $r_2$ intersect $\mathcal{P}$ at the same angle. Rotate $r_1$ around $a_1$ and rotate $r_2$ around $a_2$; denote by $\mathcal{S}_1$ and $\mathcal{S}_2$ the two surfaces of revolution they sweep out. Show that the common points of $\mathcal{S}_1$ and $\mathcal{S}_2$ lie in a plane.