Twistor lines on Nagata threefold

We give an explicit description of rational curves in the product of three copies of complex projective lines, which are transformed into twistor lines in M. Nagata's example of non-projective complete algebraic variety, viewed as the twistor space of Eguchi-Hanson metric. In particular, we show that there exist two families of such curves and both of them are parameterized by mutually diffeomorphic, connected real 4-dimensional manifolds. We also give a relationship between these two families through a birational transformation naturally associated to the Nagata's example.