Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i$ be the standard symplectic structure on $\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$. We define the following distribution $D$ on $\mathbb{R}^{2n}\setminus\{0\}$:

$D_Z=\{V\in \mathbb{R}^{2n}\mid \omega(V,Z)=0$

This is a nonintegrable distribution of codimension $1$. We define a meteic on $\mathbb{R}^{2n}\setminus\{0\}$ as follows: The distance $d(x,y)$ is the infimum of the Euclidean length of all $D$- horizontal curves joining(connecting) $x$ to $y$.

Is this metric well defoned(i.e. is this distribution totally non integrable)?Does this metric arise from a Riemmanian metric on $\mathbb{R}^{2n}\setminus\{0\}$?