I will try to clarify below the answers of Misha Verbitsky and Max Karev by reformulating then in the language of my other answer. Contrary to what I have wrote before in the comments of the main post, the distribution considered by Verbitsky is indeed polynomial.

Let $f(x,y,z,t)$ be a homogeneous polynomial and $H= f^{-1}(1)$. Consider the restriction of the $1$-form $$ \eta = \frac{\partial f}{\partial y} dx - \frac{\partial f}{\partial x} dy + \frac{\partial f}{\partial t} dz - \frac{\partial f}{\partial z} dt $$ at $H$. We want to extend the distribution defined on $H$ by it to the whole $\mathbb R^4$ in such a way that the result is invariant by homotheties.

Since $f$ does not vanish on $H$ we can multiply $\eta$ by $f$ and we will still get the same distribution on $H$. Similarly, since $df$ vanishes identically when restricted to $H$ we can sum multiples of $df$ to $\eta$ without changing the distribution on $H$. Therefore the sought homogeneous $1$-form is $$ \omega = \deg(f) f \eta - (\eta(R)) df , $$ where $R$ is Euler's vector field. Its restriction to $H$ defines the very same distribution as $\eta$, and it defines a polynomial distribution on $\mathbb P^3_{\mathbb R}$ of degree $2\deg(f) - 2$.

I will try to clarify below the answers of Misha Verbitsky and Max Karev by reformulating then in the language of my other answer. Contrary to what I have wrote before in the comments of the main post, the distribution considered by Verbitsky is indeed polynomial.

Let $f(x,y,z,t)$ be a homogeneous polynomial and $H= f^{-1}(1)$. Consider the restriction of the $1$-form $$ \eta = \frac{\partial f}{\partial y} dx - \frac{\partial f}{\partial x} dy + \frac{\partial f}{\partial t} dz - \frac{\partial f}{\partial z} dt $$ at $H$. We want to extend the distribution defined on $H$ by it to the whole $\mathbb R^4$ in such a way that the result is invariant by homotheties.

Since $f$ does not vanish on $H$ we can multiply $\eta$ by $f$ and we will still get the same distribution on $H$. Similarly, since $df$ vanishes identically when restricted to $H$ we can sum multiples of $df$ to $\eta$ without changing the distribution on $H$. Therefore the sought homogeneous $1$-form is $$ \omega = \deg(f) f \eta - (\eta(R)) df , $$ where $R$ is Euler's vector field. Notice that $\omega$ defines a section of $\Omega_{\mathbb P^3}(2d)$. Its restriction to $H$ defines the very same distribution as $\eta$, and if $Z$ stands for the divisorial components of its zero set then $\omega$ defines a polynomial distribution on $\mathbb P^3_{\mathbb R}$ of degree $2\deg(f) - 2 - \deg(Z)$.

I will try to clarify below the answers of Misha Verbitsky and Max Karev by reformulating then in the language of my other answer. Contrary to what I have wrote before in the comments of the main post, the distribution considered by Verbitsky is indeed polynomial.

Let $f(x,y,z,t)$ be a homogeneous polynomial and $H= f^{-1}(1)$. Consider the restriction of the $1$-form $$ \eta = \frac{\partial f}{\partial y} dx - \frac{\partial f}{\partial x} dy + \frac{\partial f}{\partial t} dz - \frac{\partial f}{\partial z} dt $$ at $H$. We want to extend the distribution defined on $H$ by it to the whole $\mathbb R^4$ in such a way that the result is invariant by homotheties.

Since $f$ does not vanish on $H$ we can multiply $\eta$ by $f$ and we will still get the same distribution on $H$. Similarly, since $df$ vanishes identically when restricted to $H$ we can sum multiples of $df$ to $\eta$ without changing the distribution on $H$. Therefore the sought homogeneous $1$-form is $$ \omega = \deg(f) f \eta - (i_R \eta) df , $$ where $R$ is Euler's vector field. Its restriction to $H$ defines the very same distribution as $\eta$, and it defines a polynomial distribution on $\mathbb P^3_{\mathbb R}$ of degree $2\deg(f) - 2$.

I will try to clarify below the answers of Misha Verbitsky and Max Karev by reformulating then in the language of my other answer. Contrary to what I have wrote before in the comments of the main post, the distribution considered by Verbitsky is indeed polynomial.

Let $f(x,y,z,t)$ be a homogeneous polynomial and $H= f^{-1}(1)$. Consider the restriction of the $1$-form $$ \eta = \frac{\partial f}{\partial y} dx - \frac{\partial f}{\partial x} dy + \frac{\partial f}{\partial t} dz - \frac{\partial f}{\partial z} dt $$ at $H$. We want to extend the distribution defined on $H$ by it to the whole $\mathbb R^4$ in such a way that the result is invariant by homotheties.

Since $f$ does not vanish on $H$ we can multiply $\eta$ by $f$ and we will still get the same distribution on $H$. Similarly, since $df$ vanishes identically when restricted to $H$ we can sum multiples of $df$ to $\eta$ without changing the distribution on $H$. Therefore the sought homogeneous $1$-form is $$ \omega = \deg(f) f \eta - (\eta(R)) df , $$ where $R$ is Euler's vector field. Its restriction to $H$ defines the very same distribution as $\eta$, and it defines a polynomial distribution on $\mathbb P^3_{\mathbb R}$ of degree $2\deg(f) - 2$.