  Let $V$ be a plane smooth cubic curve over a finitely generated field $k.$
The Mordell-Weil theorem for $V$ states that there is a finite subset $P\subset
V(k)$ such that the whole $V(k)$ can be obtained from $P$ by drawing secants
and tangents through pairs of previously constructed points and consecutively
adding their new intersection points with $V.$ Equivalently, the group of
birational transformations of $V$ generated by reflections with respect to
$k$-points is finitely generated. In this paper, elaborating an idea from [M3],
we establish a Mordell-Weil type finite generation result for some birationally
trivial cubic surfaces $W$. To the contrary, we prove that the birational
automorphism group generated by reflections cannot be finitely generated if
$W(k)$ is infinite.
