Pseudoholomorphic Curves in Nearly Kahler ${\mathbf{C}\mathbf{P}}^3$

We study pseudoholomorphic curves in the nearly Kalher ${\mathbf{C}\mathbf{P}}^3$. 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 ${\mathbb{C}\mathbb{P}}^3$ 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.