  For a generalized Weyl Poisson algebra $A$, explicit sets of generators and
defining relations are presented for its Poisson enveloping algebra $\CU (A)$.
Simplicity criteria are given for the algebra $\CU (A)$ and algebra of Poisson
differential operators $P\CD (A)$ on $A$. The Gelfand-Kirillov dimensions of
the algebras $\CU (A)$ and $P\CD (A)$
  are calculated. It is proven that the algebra $\CU (A)$ is a domain provided
that the coefficient ring $D$ of the generalized Weyl Poisson algebra $A$ is a
domain of essentially finite type over a perfect field.
  For the algebra $A$, the set of its minimal primes and the prime radical are
described and an equidimensionality criterion is given.
  For the equidimensional algebra $A$ of essentially finite type, two
regularity criteria are presented.
