In space, there are two non-parallel planes: $\mathbf{S}$ and $\mathbf{S}_{2}$, as well as two parallel lines: $a \| b$ and a point $P$. Construct a line $e$ passing through point $P$ that forms equal angles with the given planes and is equidistant from the given lines. (It is sufficient to submit a description of the spatial solution.)
1.) The geometric locus of lines in space that form equal angles with the given planes $\mathbf{S}_{1}, \mathbf{S}_{2}$ consists of all lines that are parallel to one of the two angle bisector planes of $\mathbf{S}_{1}$ and $\mathbf{S}_{2}$.

2.) The geometric locus of lines in space that are equidistant from the parallel lines $a$ and $b$ consists, on the one hand, of all lines that intersect the line $f$, the bisector of the distance between $a$ and $b$, at infinity, or are parallel to $a$ and $b$ and lie in the plane $\mathbf{F}$ that perpendicularly bisects the distance between $a$ and $b$; on the other hand, of all lines that are parallel to the plane $[a, b]$ but not parallel to $a$ and $b$.

Let $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ be the planes through point $P$ that are parallel to the angle bisector planes of $\mathbf{S}_{1}$ and $\mathbf{S}_{2}$. Let $[P f]=\mathbf{U}_{1}$ and the plane through $P$ that is parallel to the plane $[a, b]$ be $\mathbf{U}_{2}$. Then the lines

$$
\left|\mathbf{F}_{1} \mathbf{U}_{1}\right|, \quad\left|\mathbf{F}_{1} \mathbf{U}_{2}\right|, \quad\left|\mathbf{F}_{2} \mathbf{U}_{1}\right| \quad \text { and } \quad\left|\mathbf{F}_{2} \mathbf{U}_{2}\right|
$$

are the sought lines, provided they are not parallel to $a$ and $b$ without coinciding with $\mathbf{F}$.

In general, we have thus found 4 lines that satisfy our conditions.

If the intersection line of $\left|\mathbf{S}_{1} \mathbf{S}_{2}\right|$ is parallel to $a$ and $b$, then the 4 lines coincide with one line parallel to $a$ and $b$, and form only 1 solution if $P$ lies in the plane $\mathbf{F}$.

If among the 4 pairs of planes, each of whose intersection lines provides 1 solution, there is a coincident pair (e.g., $\mathbf{F}_{1} \equiv \mathbf{U}_{2}$), then instead of one solution, we obtain infinitely many solutions (a pencil of lines), without this condition affecting the other solutions different from this pencil.