Suppose you have smooth foliation on a Euclidean ball $\mathbb{B}^{4} \subset \mathbb{C}^{2}$, whose leaves are holomorphic curves with respect to the standard complex structure. Let $(z_{1},z_{2})$ be coordinates on $\mathbb{C}^{2}$.
Suppose that at every point $p$ in the $z_{1}$-axis, the leaf of the foliation through $p$ meets the $z_{1}$-axis tangentially.
Can we deduce that in fact the $z_{1}$-axis must be a leaf of the foliation? how to show it?
If you take a submersion $g$ defining the foliation around $p \in \{ z_1 =0 \}$ and compose with the inclusion $h\colon \mathbb{D} \rightarrow \{ z_1 =0 \} \subset\mathbb{C}^2$. The tangency implies that $g\circ h$ is constant, hence $g$ is constant on $\{ z_1 =0 \}$. The result follows by connectedness.