  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.
