Calibrated Submanifolds of R⁷ and R⁸ with Symmetries

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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 R⁷ and SU(2)-invariant Cayley 4-folds in R⁸ 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 R⁷ and R⁸ are given.