The Kontsevich tetrahedral flow in 2D: a toy model

In the paper “Formality conjecture” (1996) Kontsevich designed a universal flow $\dot{\mathcal{P}}={\mathcal{Q}}_{a:b}(\mathcal{P})=a{\mathrm{\Gamma }}_1+b{\mathrm{\Gamma }}_2$ on the spaces of Poisson structures $\mathcal{P}$ on all affine manifolds of dimension $n⩾2$. We prove a claim from $\mathit{\text{loc.}}\mathit{\text{cit.}}$ stating that if $n=2$, the flow ${\mathcal{Q}}_{1:0}={\mathrm{\Gamma }}_1(\mathcal{P})$ is Poisson-cohomology trivial: ${\mathrm{\Gamma }}_1(\mathcal{P})$ is the Schouten bracket of $\mathcal{P}$ with $\mathcal{X}$, for some vector field $\mathcal{X}$; we examine the structure of the space of solutions $\mathcal{X}$. Both the construction of differential polynomials ${\mathrm{\Gamma }}_1(\mathcal{P})$ and ${\mathrm{\Gamma }}_2(\mathcal{P})$ and the technique to study them remain valid in higher dimensions $n⩾3$, but neither the trivializing vector field $\mathcal{X}$ nor the setting $b:=0$ survive at $n⩾3$, where the balance is $a:b=1:6$.