Extrinsic geometry and linear differential equations of ${\mathfrak{s}\mathfrak{l}}_3$-type

As an application of the general theory on extrinsic geometry, we investigate extrinsic geometry in frag varieties and systems of linear PDE’s for a class of special interest associated with the adjoint representation of $\mathfrak{s}\mathfrak{l}(3)$. We carry out a complete local classification of the homogeneous structures in this class. As a result, we find 7 kinds of new systems of linear PDE’s of second order on a 3-dimensional contact manifold each of which has a solution space of dimension 8. Among them there are included a system of PDE’s called contact Cayley’s surface and one which has $\mathfrak{s}\mathfrak{l}(2)$ symmetry.