2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77086We investigate, in some details, symplectic equivalence between several conformal classes of Lorentz metrics on the hyperboloid of one sheet $H^{1,1} \cong T \times T - Δ$ and affine coadjoint orbits of the group $Diff_+(Δ)$ of orientation preserving diffeomorphisms of $Δ\cong T$ with its natural projective structure. This will allow for generalizations, namely, to the case of arbitrary projective structures on null infinity.27 pages, LaTeXDifferential GeometrySymplectic Geometrytext