On pairs, triples and quadruples of points on a cubic surface

Let $X^{(n)}$ denote $n$-th symmetric power of a cubic surface $X$. We show that $X^{(4)}\times X$ is stably birational to $X^{(3)}\times X$, despite examples when $X^{(4)}$ is not stably birational to $X^{(3)}$.
Comment: English and Russian versions, 2 pages each. v1: Article in Russian, v2: English translation, v3: added Afterword on page 2 (only in English). Comments welcome!