2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/156094The principal theory of this paper comprises a technique for constructing associative, coassociative and Cayley submanifolds of Euclidean space with symmetries, using first-order ordinary differential equations. Explicit examples of U(1)-invariant associative cones in R^7 and SU(2)-invariant Cayley 4-folds in R^8 are then produced using this method. Further examples of associative 3-folds are presented, which are ruled, and other systems of differential equations defining calibrated submanifolds in R^7 and R^8 are given.21 pages, minor corrections, section added containing further systems of differential equations defining calibrated submanifoldsDifferential Geometry53C38text