Weighted Poisson polynomial rings in dimension three

We discuss Poisson structures on a weighted polynomial algebra $A:=\Bbbk[x, y, z]$ defined by a homogeneous element $Ω\in A$, called a potential. We start with classifying potentials $Ω$ of degree deg$(x)+$deg$(y)+$deg$(z)$ with any positive weight (deg$(x)$, deg$(y)$, deg$(z)$) and list all with is