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  1. We study Poisson structures on a weighted polynomial algebra A of three variables defined by a homogeneous potential. We start with classifying potentials w 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/(w) for irreducible potentials w. Using Poisson valuations, we characterize the Poisson automorphism group of A when it 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. 
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