Given a smooth complex structure $X$ of algebraic dimension $0$ on the $6$-sphere $S^6$, we study various geometric aspects of $X$. We show that $X$ cannot be covered by a compact family of curves, hence does not admit any meromorphic dominant map to a surface. In case $X$ does not carry any curve, either the tangent bundle or some quotient of rank $2$ has a Hermite-Einstein metric. We study further the algebraic dimension of the projectivized tangent bundle $\mathbb{P}(T_X)$ as well as the geometry of surfaces in $X$. Finally, we construct new manifolds out of $X$ (in the hope that they might lead to a contradiction to the existence of $X$).