  As is known to all, Hopf-Galois objects have a significant research value for
analyzing tensor categories of comodules and classification questions of
pointed Hopf algebras, and are natural generalizations of Hopf algebras with a
Galois-theoretic flavour. In this paper, we mainly prove a criterion for an Ore
extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, and introduce
the conception of Poisson Hopf-Galois algebras, and establish the relationship
between Poisson Hopf-Galois algebras and Poisson Hopf algebras. Moreover, we
study Poisson Hopf-Galois structures on Poisson polynomial algebras, and mainly
give a necessary and sufficient condition for the Poisson enveloping algebra of
a Poisson Hopf-Galois algebra to be a Hopf-Galois algebra.
