Let $(M,g)$ be a Riemannian manifold which admit a non vanishing vector field.(That is $\chi(M)=0$ when $M$ is a compact manifold). We pull back The symplectic structure of the cotangent bundle to the $2$-form $\omega$ on $TM$.

Is there necessarily a non vanishing vector field $X$ on $M$ for which the following submanifold of $(TM, \omega)$ would be a symplectic submanifold?

$$\{v_p\in TM \mid |v_p|=1, v \perp X(p)\}$$

where $v_p$ is a vector in $TM$ based at point $p\in M$.

The motivation for this question is the following:

We would like to find some symplectic submanifolds of $TM$ which are in the form of a sub vector bundle of the tangent bundle or sub fiber bundle of unite tangent bundle.

In the standard coordinate $(x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n)$, the elementary examples of symplectic submanifolds are $$(x_1,x_2,\ldots,x_k,0,0,\ldots,0,y_1,y_2,\ldots,y_k,0,0\ldots,0)$$

In such elementary example we loose the whole base space.