The Killing equation and its generalisations for Killing tensors are important for the study of symmetry, structure deformations, and for yielding integrable solutions of Hamiltion-Jacobi and Laplacian equations. These Killing equations arise in some differential complexes known as BGG complexes. Understanding this is one of the key steps to a new approach to constructing collections of solutions to these equations that we outline in this talk. The BGG complex knowledge enables easy access to the PDE prolongation. The interaction of the prolonged system with an object known as the scale tractor yields the desired solutions, which then provide first integrals for equations such as the geodesic equation, other Hamiliton-Jacobi equations and for Schrodinger operators equations. The overall story demonstrates a beautiful interaction of mechanics, geometry, and
algebra.
This is joint work with Simon Goodwin, Thomas Leistner, and Jonathan Kress.