2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/182013We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one.22 pages. v2: remarks added concerning the parameter m having no effect on the metric, and the non-existence of metrics on the link L(2,2,2,3); statements and proofs of main theorems modified in light of a more precise characterization of the cohomogeneity one diagrams. v3: presentation improved. To appear in Comm. Math. PhysDifferential GeometryHigh Energy Physics - Theory53C25 (Primary) 53C30, 57S15 (Secondary)text