Let us consider polynomial contact structures on $\mathbb RP^3$, i.e. contact structures on $\mathbb R^3$ defined by a form $w=Pdx+Qdy+Rdz,\ P,Q,R\in \mathbb R[x,y,z]\ $ in an affine part and then extended to $\mathbb RP^3$, and $ w \wedge dw \ne 0$ everywhere.

One can find all such forms $w$ that $deg P, deg Q, deg R \leq 1$ by direct calculation:

$w=(qy-rz+a)dx+ (pz-qx+b)dy + (rx-py+c)dz,\ a,b,c,p,q,r\in \mathbb R;\ ap+br+cq \ne 0$.

But I can't do anything for greater degrees. Do you know any criteria for coefficients of $P,Q,R$?

Does anybody know any contact polynomial form with $deg P, deg Q, deg К \leq 2$?

Added: What is the form (I mean form coefficients in $\mathbb R^3\subset \mathbb RP^3$) defines polynomial contact structure constructed by plurisubharmonic function $f=x^4+y^4+z^4+t^4$?