  We consider self-dual metrics on 3CP^2 of positive scalar curvature admitting
a non-trivial Killing field, but which is not conformally isometric to LeBrun's
metrics. Firstly, we determine defining equations of the twistor spaces of such
self-dual metrics. Next we prove that conversely, the complex threefolds
defined by the equations always become twistor spaces of self-dual metrics on
3CP^2 of the above kind. As a corollary, we determine a global structure of the
moduli spaces of these self-dual metrics; namely we show that the moduli space
is non-empty and isomorphic to R^3/G, where G is an involution of R^3 having
one-dimensional fixed locus. Combined with works of LeBrun, this settles a
moduli problem of self-dual metrics on 3CP^2 of positive scalar curvature
admitting a non-trivial Killing field. In our proof, a key role is played by a
classical result in algebraic geometry that a smooth plane quartic always
possesses twenty-eight bitangents.
