“…It what follows we shall classify all Poisson algebras Q that admit a surjective Poisson algebra map Q → P → 0 with a 1-dimensional kernel. In order to do this we first recall from [3] the following concept: (2) Two Poisson algebras P (λ, Λ, ϑ, γ, f ) and P (λ ′ , Λ ′ , ϑ ′ , γ ′ , f ′ ) are isomorphic if and only if there exists a triple (s 0 , ψ, r) ∈ k * ×Aut Poss (P )×Hom k (P, k) such that for any p, q ∈ P : which is precisely (19). The first part is finished once we observe that the map given by the formula (21), namely ϕ : P (λ, u) → P × k, ϕ(p, x) := (p, λ(p) + u x), for all p ∈ P and x ∈ k is an isomorphism of Poisson algebras.…”