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For a family of contact forms, require $Y_t\in \ker {\alpha }_t$ and solve $({\iota }_{Y_t}d{\alpha }_t){|}_{\ker {\alpha }_t}=-{\dot{\alpha }}_t{|}_{\ker {\alpha }_t}$. The contact distribution symplectic form gives a unique smooth solution. Then ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, with $h_t={\dot{\alpha }}_t(R_t)$ and $R_t$ the Reeb vector field.

Put ${\xi }_t=\ker {\alpha }_t$, the contact distribution. If $\dim M=2r+1$, the contact form condition is ${\alpha }_t\wedge (d{\alpha }_t{)}^r\ne 0$. It implies that $d{\alpha }_t{|}_{{\xi }_t}$ is a symplectic form on each contact hyperplane: inserting a vector transverse to ${\xi }_t$ in that volume form leaves the nonzero top exterior power $(d{\alpha }_t{|}_{{\xi }_t}{)}^r$. Hence there is a unique smooth time-dependent vector field $Y_t\in {\xi }_t$ satisfyingIts smoothness follows by inverting the smoothly varying nondegenerate matrix of this contact distribution symplectic form. This is the horizontal generator of contact stability.

Let $R_t$ be the Reeb vector field of ${\alpha }_t$. The one-form ${\dot{\alpha }}_t+{\iota }_{Y_t}d{\alpha }_t$ vanishes on ${\xi }_t$, so it is $h_t{\alpha }_t$. Evaluating on $R_t$, using ${\alpha }_t(R_t)=1$ and ${\iota }_{R_t}d{\alpha }_t=0$, identifies the coefficient:By Cartan's magic formula, the Lie derivative of a differential form isbecause $Y_t$ lies in the contact hyperplane.

Let ${\rho }_t$ solve ${\partial }_t{\rho }_t=Y_t\circ {\rho }_t$, with ${\rho }_0={\mathrm{id}}_M$. Since $M$ is compact without boundary and $Y_t$ is smooth on the closed time interval, its solutions cannot escape and exist for the entire interval. Reversing the time-dependent equation supplies a smooth inverse for each ${\rho }_t$. Thus these flow maps give a smooth isotopy. Differentiating the pullback of a differential form along this isotopy yieldsFor each base point this is a scalar linear equation for a covector, initially ${\alpha }_0$. Its solution isBy finite-time flow completeness on a compact manifold, this construction gives the whole time interval. The contact conformal factor along an isotopy is smooth in $(t,x)$ and nowhere zero, proving Gray stability theorem. The precise domains here are ${\rho }_t:M\to M$ and $\rho :M\times [0,1]\to M$; the extra time factor attached to the already indexed ${\rho }_t$ in the source is a notational slip. The proof produces the required family on its whole specified interval. If a parameter domain of all $\mathbb{R}$ is intended, extend the smooth field slightly beyond $[0,1]$, multiply it by a time cutoff, and use compactness to obtain the extended isotopy $\rho :M\times \mathbb{R}\to M$. Its restriction to $[0,1]$ is unchanged.

Use polar coordinates $z_j=r_j{e}^{i{\theta }_j}$ on each complex coordinate plane. On $S^3$, where $r_1^2+r_2^2=1$,Direct substitution shows that ${\alpha }_s\wedge d{\alpha }_s$ is nowhere zero for $s>0$, so ${\alpha }_s$ is a contact form.

The vector fieldsatisfies ${\alpha }_s(R_s)=r_1^2+r_2^2=1$ and ${\iota }_{R_s}d{\alpha }_s=0$ on tangent vectors to the sphere. It is therefore the Reeb vector field. Its flow isIf both coordinates are nonzero, an orbit closes only if the two angular frequencies have rational ratio, equivalently if $s\in \mathbb{Q}$. For irrational $s$, the only closed orbits are $\{z_2=0\}$ and $\{z_1=0\}$, the two coordinate circles. This is the irrational contact ellipsoid flow on the three-sphere.