  We discuss Poisson structures on a weighted polynomial algebra $A:=\Bbbk[x,
y, z]$ defined by a homogeneous element $\Omega\in A$, called a potential. We
start with classifying potentials $\Omega$ of degree deg$(x)+$deg$(y)+$deg$(z)$
with any positive weight (deg$(x)$, deg$(y)$, deg$(z)$) and list all with
isolated singularity. Based on the classification, we study the rigidity of $A$
in terms of graded twistings and classify Poisson fraction fields of
$A/(\Omega)$ for irreducible potentials. Using Poisson valuations, we
characterize the Poisson automorphism group of $A$ when $\Omega$ has an
isolated singularity extending a nice result of
Makar-Limanov-Turusbekova-Umirbaev. Finally, Poisson cohomology groups are
computed for new classes of Poisson polynomial algebras.
