  In this article, we use the harmonic sequence associated to a weakly
conformal harmonic map $f:S\to S^6$ in order to determine explicit examples of
linearly full almost complex 2-spheres of $S^6$ with at most two singularities.
We prove that the singularity type of these almost complex 2-spheres has an
extra symmetry and this allows us to determine the moduli space of such curves
with suitably small area. We also characterize projectively equivalent almost
complex curves of $S^6$ in terms of $G_2^{\cc}$-equivalence of their directrix
curves.
