The Kontsevich tetrahedral flow in 2D: a toy model

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In the paper "Formality conjecture" (1996) Kontsevich designed a universal flow P=Qₐ:b(P)=aΓ₁+bΓ₂ on the spaces of Poisson structures P on all affine manifolds of dimension n 2. We prove a claim from loc. cit. stating that if n=2, the flow Q₁:₀=Γ₁(P) is Poisson-cohomology trivial: Γ₁(P) is the Schouten bracket of P with X, for some vector field X; we examine the structure of the space of solutions X. Both the construction of differential polynomials Γ₁(P) and Γ₂(P) and the technique to study them remain valid in higher dimensions n 3, but neither the trivializing vector field X nor the setting b:=0 survive at n 3, where the balance is a:b=1:6.