Integrability of $\ker \alpha(X,.)$

Question

Let $N^3$ and $M^4$ be Riemannian Manifolds such that $N$ is a submanifold of $M^4$ and $\alpha$ be the second fundamental form of $N$. Given a nowhere-vanishing vector field $X \in TN$, is it true that $\ker \alpha(X,.)$ is integrable? If not, under what conditions is it possible to state that $\ker \alpha(X,.)$ is integrable?