Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R P}^2$-structures in genus one

In this paper we study a broad class of complete Hamiltonian in-tegrable systems, namely the ones whose associated Lagrangian fibration is complete and has non compact fibres. By studying the associated complete Lagrangian fibration, we show that, under suitable assumptions, the integrals of motion can be taken as action coordinates for the Hamiltonian system. As an application we find global Darboux coordinates for a new family of symplectic forms ωf, parametrized by smooth functions f: [0, +∞) → (−∞, 0], defined on the deformation space ofIrh3OF4bEy6/oxLVHc6EWAKkHAOzproperly convex RP2-structures on the torus. Such a symplectic form is part of a family of pseudo-Kähler metrics (gf, I, ωf) defined on B0 (T2) and introduced by the authors. In the last part of the paper, by choosing f(t) = −kt, k > 0 we deduce the expression for an arbitrary isometry of the space.