Explicit solutions of PDE via Vessiot theory and solvable structures

We consider the problem of computing the integrable sub-distributions of the non-integrable Vessiot distribution of multi-dimensional second order partial differential equations (PDEs). We use Vessiot theory and solvable structures to find the largest integrable distributions contained in the Vessiot distribution associated to second order PDEs. In particular, we show how the solvable symmetry structure of the original PDE can be used to construct integrable sub-distributions leading to group invariant solutions of the PDE in two and more than two independent variables.