I) Consider a Riemannian 3-manifold $(M,g)$ with a non-vanishing vector field $V\in \Gamma(TM)$. Define a 2-dimensional distribution as
$$\tag{1} \Delta ~:=~ {\rm ran} (V)^{\perp}~\subseteq~ TM.$$
This definition (1) and the integrability condition
$$\tag{2} V \cdot \nabla\times V ~=~ 0$$
for $\Delta$ relies on a choice of dot product/metric tensor $g$.
II) A more fundamental approach (that doesn't depend of a choice of metric) is
the following. Consider a 3-manifold $M$ with a non-vanishing$^1$ co-vector field/one-form $\vartheta\in \Gamma(T^{\ast}M)$. Define a 2-dimensional distribution as
$$\tag{3} \Delta ~:=~ {\rm ker}(\vartheta)~\subseteq~ TM.$$
The integrability condition for $\Delta$ reads
$$\tag{4} \vartheta \wedge \omega ~=~ 0 ,\qquad \omega~:=~\mathrm{d}\vartheta.$$
We will use eqs. (3)-(4) from now on [as opposed to eqs. (1)-(2)].
III) The integrability condition (4) is equivalent to
$$\tag{5} \exists \text{ locally defined gauge potential } A:~~ D\vartheta ~:=~ \mathrm{d}\vartheta -A \wedge \vartheta~=~0. $$
Note that eqs. (3)-(5) are covariant under a gauge transformation
$$\tag{6} \vartheta~\longrightarrow ~ \vartheta^{\prime}~=~e^{f}\vartheta, \qquad A ~\longrightarrow ~A^{\prime}~=~A+ \mathrm{d} f, $$
where $f\in C^{\infty}(M)$ is a gauge function. Eq. (5) is also covariant under
a shift symmetry
$$\tag{7} A~\longrightarrow ~A^{\prime}~=~A+ g\vartheta, $$
where $g\in C^{\infty}(M)$ is a function.
IV) One may show that the integrability condition (4) is equivalent to the involutive property of vector fields
$$\tag{8} \forall X,Y \text{ vector fields }\in~\Delta: ~~[X,Y]~\in~\Delta, $$
cf. Frobenius' theorem.
V) Moreover the integrability condition (4) implies that there is a 2-dimensional foliation of $M$, i.e. there exists an atlas of adapted local coordinate systems $(x,y,z)$ so that the third-component $\phi^3$ of a local coordinate transformation
$$\tag{9}(x,y,z) ~\longrightarrow~ (x^{\prime},y^{\prime},z^{\prime})~=~(\phi^1(x,y,z),\phi^2(x,y,z),\phi^3(z)) $$
only depends on the third coordinate $z$. In these adapted coordinate systems, we have
$$\tag{10} \frac{\partial}{\partial x}, \frac{\partial}{\partial y} ~\in~\Delta,\qquad \text{and} \qquad\vartheta~\propto~\mathrm{d}z.$$
VI) Further equivalent formulations of the integrability condition (4):
$$\tag{11} \exists \text{ local integrating factor } e^f:~~ e^f\vartheta \text{ is closed}. $$
Equivalently, there exists a local gauge (6), where $\Delta={\rm ker}(\vartheta^{\prime})$, such that$^2$
$$\tag{12} \vartheta^{\prime}\text{ is closed}. $$
By Poincare lemma, we have
$$\tag{13} \exists \text{ local coordinate system } (x,y,z) :~~ \vartheta^{\prime}~=~\mathrm{d}z.$$
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$^1$ If we instead of imposing that the 2-form $\omega:=\mathrm{d}\vartheta$ is non-vanishing (rather than the 1-form $\vartheta$ is non-vanishing), then $\omega$ must have constant rank 2, and we can use the Darboux' Theorem to conclude that
$$\tag{14} \exists \text{ local coordinate system } (x,y,z) :~~ \vartheta~=~y\mathrm{d}z.$$
$^2$ In particular, there exists one-forms $\vartheta$ in 3D which doesn't have an integrating factor.
