Let $(M^{n+1}, \langle \cdot, \cdot \rangle)$ be a parallelizable Riemannian manifold with a vector bundle isomorphism
$$\varphi : TM \to M \times \mathbb{R}^{n+1}.$$
For $x \in M$, denote by $\varphi_x : T_x M \to \mathbb{R}^{n+1}$ the restriction of $\varphi$ to the tangent space $T_x M$; it is a linear isomorphism.
Given a vector $v \in \mathbb{R}^{n+1}$, define a vector field $V = V_v$ on $M$ by
$$V(x) = \varphi_x^{-1}(v), \quad x \in M.$$
Question: when is the normal distribution associated to $V$ integrable? That is, if $\mathcal{D} = (\mathcal{D}_x)_{x \in M}$ is defined by $\mathcal{D}_x = \{ V(x) \}^{\perp}$, when is $\mathcal{D}$ integrable? What conditions can we impose on $\varphi$ for this to be true?