Sp(3,R) Monge geometries in dimension 8

We study a geometry associated with rank 3 distributions in dimension 8, whose symbol algebra is constant and has a simple Lie algebra sp(3,R) as Tanaka prolongation. We restrict our considerations to only those distributions that are defined in terms of a systems of ODEs of the form z˙ij=∂2f(x˙1,x˙2)∂x˙i∂x˙j, i≤j=1,2. For them we built the full system of local differential invariants, by solving an equivalence problem à la Cartan, in the spirit of his 1910's five variable paper. The considered geometry is a parabolic geometry, and we show that its main invariant - the harmonic curvature - is a certain quintic. In the case when this quintic is maximally degenerate but nonzero, we use Cartan's reduction procedure and reduce the EDS governing the invariants to 11, 10 and 9 dimensions. As a byproduct all homogeneous models having maximally degenerate harmonic curvature quintic are found. They have symmetry algebras of dimension 11 (a unique structure), 10 (a 1-parameter family of nonequivalent structures) or 9 (precisely two nonequivalent structures).