  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
$\mathfrak{sl}(3)$. 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
$\mathfrak{sl}(2)$ symmetry.
