  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.
