  We compute the formal Poisson cohomology groups of a real Poisson structure
$\pi$ on $\mathbb{C}^2$ associated to the Lefschetz singularity $(z_1,
z_2)\mapsto z_1^2+z_2^2$. In particular we correct an erroneous computation in
the literature. The definition of $\pi$ depends on a choice of volume form.
Using the main result we formally classify all Poisson structure arising from
different choices of volume forms.
