Partial extensions of jets and the polar distribution on Grassmannians of non-maximal integral elements

We study an intrinsic distribution, called polar, on the space of $l$-dimensional integral elements of the higher order contact structure on jet spaces. The main result establishes that this exterior differential system is the prolongation of a natural system of PDEs, named pasting conditions, on sections of the bundle of partial jet extensions. Informally, a partial jet extension is a $k$th order jet with additional $k+1$st order information along $l$ of the $n$ possible directions. A choice of partial extensions of a jet into all possible $l$-directions satisfies the pasting conditions if the extensions coincide along pairwise intersecting $l$-directions. We further show that prolonging the polar distribution once more yields the space of $(l,n)$-dimensional integral flags with its double fibration distribution. When $l>1$ the exterior differential system is holonomic, stabilizing after one further prolongation. The proof starts form the space of integral flags, constructing the tower of prolongations by reduction.