  Let $P = \Bbbk[x1,x2,x3]$ 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 $\widetilde{G}$.
