  The 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.
