2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222638We show that any hyperbolic Inoue surface (or Inoue-Hirzebruch surface of even type) admits anti-self-dual bihermitian structures. The same result also holds for any of its small deformations as far as its anti-canonical system is non-empty. Similar results are obtained for parabolic Inoue surfaces. Our method also yields anti-self-dual hermitian metrics on any half Inoue surface. We use the twistor method of Donaldson-Friedman for the proof.69 pages,Differential GeometryAlgebraic Geometry53C55 (Primary) 53C28, 32G05 (Secondary)text