Invariants of Unimodular Quadratic Polynomial Poisson Algebras of Dimension 3

Let $P=\mathbb{k}[x1,x2,x3]$ be a unimodular quadratic Poisson algebra and let $G$ be a finite subgroup of the graded Poisson automorphism group of $P$. In this paper, we prove a variant of the Shephard-Todd-Chevalley theorem for $P$ and variants the Shephard-Todd-Chevalley theorem and the Watanabe theorem for its Poisson enveloping algebra $U(P)$ under the induced group $\tilde{G}$.