$M$ : differentiable manifold
$\omega$ : 2-form of $M$
2-form $\omega$ is defined to be nonsingular if dim$(\mathrm{Ker}\omega_x^{\flat})=1$ at any point $x \in M$, where$\omega^{\flat}_x : T_x M \rightarrow T^*_xM, v \mapsto \omega^{\flat}_x(v):= \omega_x(v,\cdot)$.
When the tangent vector $\dot{c}(t):=c_*(\frac{\partial}{\partial t})$ of a smooth curve $c : \mathbb{R}\rightarrow M, t \mapsto c(t)$ satisfies $\dot{c}( t) \neq 0$ and $\omega^{\flat}_{c(t)}(\dot{c}(t))=0$ in any $t\in \mathbb{R}$, we call $c$ a null curve in $\omega$.
My textbook said that if $M$ is an odd dimensional manifold and a 2-form $\omega$ of $M$ is nonsingular, then for any point in $M$ there exists a unique null curve of $\omega$ through that point, but the proof was omitted. Could you please tell me this proof or let me know the book in which the proof is written?
What is true is that through for any $p \in M$ and $v \in \ker \omega_p$, there exists a unique curve $c\colon \Bbb R \to M$, up-to parametrization, such that $c(0) = p$, $\dot c(0) = v$ and $\dot c(t) \in \ker \omega_{c(t)}$ for all $t$. The proof, in short, is the following: $\ker \omega$ is an integrable regular distribution. Let me expand on this.
$\newcommand{\D}{\mathcal{D}} \newcommand{\dx}{\mathrm{d}}$ Let $\D = \ker \omega \subset TM$, which is by assumption a rank $1$ tangent distribution. It is a classical thing that a rank $1$ distribution is always integrable. Here is a proof that is specific to this example. Fix $X$ and $Y$ two vector fields tangent to $\D$. Let $Z$ any third vector field and $\alpha$ be the $1$-form defined as $\alpha(V) = \omega(V,Z)$. Since $\D$ has rank $1$, $X$ and $Y$ are pointwise colinear, so that $\dx\alpha(X,Y) = 0$. Hence \begin{align} 0 &= \dx\alpha(X,Y) \\ &= X(\alpha(Y)) - Y(\alpha(X)) - \alpha([X,Y])\\ &= X(\omega(X,Z)) - Y(\omega(Y,Z)) - \omega([X,Y],Z)\\ &= -\omega([X,Y],Z). \end{align} Hence, $\omega([X,Y],Z) = 0$ for any $Z$. It follows that $[X,Y]$ is tangent to $\D$. One then concludes thanks to Frobenius theorem: $M$ is foliated by the integral submanifolds of $\D$, which are precisely unparametrized null-curves of $\omega$.