Extrinsic geometry and linear differential equations of \(\mathfrak {sl}_3\)-type

As an application of the general theory on extrinsic geometry (Doubrov et al. in SIGMA Symmetry Integr Geom Methods Appl 17:061, 2021), we investigate extrinsic geometry in flag 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 \(\varvec{\mathfrak {sl}}(2)\) symmetry.