  We study pseudoholomorphic curves in the nearly Kalher $\mathbf{CP}^3$. It is
shown that a class of curves called null-torsion are in one to one
correspondence with the integrals of a holomorphic contact system on the usual
Kahler $\mathbb{CP}^3$ studied by Bryant. Browing Bryant's result we get plenty
of such curves. Rational curves are shown to be either vertical or horizontal
or null-torsion. Both horizontal and null-torsion curves can be understood by
mentioned Bryant's results while vertical curves are just fibers of the twistor
fibration.

Differential Geometry