Explicit formulas for composition and inverse in the local symplectic groupoid model for the b^2-Poisson structure on R^2

Determine explicit formulas for the groupoid multiplication and inverse maps in the local symplectic groupoid integrating the Poisson manifold (R^2, π) with π = x^2(∂x ∧ ∂y), in the explicit local model CH ⊂ R^4 with coordinates (a,b,x,y) described by the authors, where the source and target maps are given by s(a,b,x,y) = (x,y) and t(a,b,x,y) = (x/(1 − a x), −b x^3/(1 − a x) + y), respectively. The goal is to derive closed-form expressions for the composition and inverse operations in these coordinates that satisfy the groupoid axioms and are compatible with the provided source and target maps and the local Poisson structure on CH.

Background

In Subsubsection 3.4.2, the authors provide an explicit local description of the symplectic groupoid integrating Poisson structures of the form π = f(x) ∂x ∧ ∂y on R^2, including the case f(x) = x^m. They present explicit formulas for the source and target maps and the local Poisson bivector on a neighborhood CH ⊂ R^4 of the identity bisection.

For the specific case m = 2 (the b^2-Poisson structure π = x^2 ∂x ∧ ∂y), they obtain concrete formulas t(a,b,x,y) = (x/(1 − a x), −b x^3/(1 − a x) + y) and s(a,b,x,y) = (x,y), and an explicit bivector for the groupoid. However, despite having s, t, and the Poisson structure, they note that the composition and inverse operations remain unspecified in this local chart.