2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/183282We examine integrability of self-dual Yang-Mills system in the Higgs phase, with taking simpler cases of vortices and domain walls. We show that the vortex equations and the domain-wall equations do not have Painleve property. This fact suggests that these equations are not integrable.15 pages, no figures, v2: references added, v3: typos corrected, the final version to appear in NPBHigh Energy Physics - Theorytext