  In this paper we study a broad class of complete Hamiltonian integrable
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 $\boldsymbol{\omega}_f$, parametrized by smooth functions
$f:[0,+\infty)\to(-\infty,0]$, defined on the deformation space of properly
convex $\mathbb{R}\mathbb{P}^2$-structures on the torus. Such a symplectic form
is part of a family of pseudo-K\"ahler metrics
$(\mathbf{g}_f,\mathbf{I},\boldsymbol{\omega}_f)$ defined on
$\mathcal{B}_0(T^2)$ 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.
