2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/199059We apply algebraic and vertex operator techniques to solve two dimensional reduced vacuum Einstein's equations. This leads to explicit expressions for the coefficients of metrics solutions of the vacuum equations as ratios of determinants. No quadratures are left. These formulas rely on the identification of dual pairs of vertex operators corresponding to dual metrics related by the Kramer-Neugebauer symmetry.20 pages; added comparison with Belinskii-Zakharov method and remarksExactly Solvable and Integrable SystemsGeneral Relativity and Quantum CosmologyHigh Energy Physics - Theorytext