SCHWEITZER 2018 Problem 10

In 3-dimensional hyperbolic space, we are given a plane $P$ and four distinct straight lines: the lines $a_1$ and $a_2$ are perpendicular to $P$ ; while the lines $r_1$ and $r_2$ do not intersect $P$ , and their distances from $P$ are equal. Denote by $S_i$ the surface of revolution obtained by rotating $r_i$ around $a_i$ . Show that the common points of $S_1$ and $S_2$ can be covered by two planes.