  We establish that, over certain ground fields, the set of osculating tangents
of Cayley's ruled cubic surface gives rise to a (maximal partial) spread which
is also a dual (maximal partial) spread. It is precisely the Betten-Walker
spreads that allow for this construction. Every infinite Betten-Walker spread
is not an algebraic set of lines, but it turns into such a set by adding just
one pencil of lines.
