Linear degeneracy of a PDE is a concept that is related to a number of interesting geometric constructions. We first take a quadratic line complex, which is a three parameter
family of lines in projective space P3 specified by a single quadratic relation in the Plucker coordinates. This complex supplies us with a conformal structure in P3.
With this conformal structure, we associate a three-dimensional second order quasilinear wave equation. We show that any PDE arising in this way is linearly degenerate, furthermore, any linearly degenerate PDE can be obtained by this construction. We
classify Segre types of quadratic complexes for which the structure is conformally flat, as well as Segre types for which the corresponding PDE is integrable. These results were published in . We then introduce the notion of characteristic integrals, discuss characteristic integrals in 3D and show that, for certain classes of second-order linearly degenerate dispersionless integrable PDEs, the corresponding characteristic integrals are parameterised by points on the Veronese variety. These results were published in .
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.