  If $\Delta$ and $\Gamma$ are two derivations of a commutative algebra $A$
such that $\Delta\Gamma-\Gamma\Delta=\Delta$ is locally nilpotent, one can
endow $A$ with a new product $\ast$ whose filtered semiclassical limit is the
Poisson structure $\Delta\wedge\Gamma$.In this article we first study theses
(Poisson) algebras from an algebraic point of view, and when $A$ is a
polynomial algebra, we investigate their homological properties. In particular,
when the derivations $\Delta$ and $\Gamma$ are linear, the algebras $(A,\ast)$
provide, in each dimension at least four, new examples of multiparameter
families of Artin-Schelter regular algebras. These algebras are deformations of
Poisson algebras $(A,\Delta\wedge\Gamma)$ of rank $2$, thus explaining the
title of the article.Assuming furthermore a technical condition on $\Gamma$, we
show that the algebra $(A,\ast)$ is Calabi-Yau if and only if the trace of
$\Gamma$ is equal to $1$ if and only if the Poisson algebra
$(A,\Delta\wedge\Gamma)$ is unimodular.Since the trace of $\Gamma$ is a linear
function of the parameters, the algebras $(A,\ast)$ also provide, in each
dimension at least four, new examples of multiparameter families of Calabi-Yau
algebras.
