This answer comes from Sheng-Fu Chiu at Northwestern.

Let $$\alpha = dz - ydx$$ be the standard contact form on $\mathbb R^3$. Let $a = z/x - y$. Let $\tan(\theta)$ be the unique real solution of $T^3 - aT^2 + 3T - a = 0,$ with branch chosen so that $\cos(\theta)$ and $x$ have the same sign.

Put $r = x/\cos(\theta), t = y + 3\sin(\theta)\cos(\theta),$ and set $R = \log(r).$ These definitions invert the coordinate transformation $$(x,y,z) = (rc,t-3sc,rs^3 + trc),$$ where $c = \cos(\theta)$ and $s = \sin(\theta)$ and $r>0$.

Put $\rho = \sqrt{(3s-2s^3)^2 + c^2}$ and define $\phi$ by $\cos(\phi) = (3s-2s^3)/\rho, \sin(\phi) = c/\rho.$ Then $$\alpha = \frac{e^R}{\rho}(\cos(\phi)dR + \sin(\phi)dt),$$ which defines the standard contact structure on the cosphere bundle of the $(R,t)$ plane.