  Let $Q$ be a quiver of extended Dynkin type $D$. In this first of two papers,
we show that the quiver Grassmannian $Gr_e(M)$ has a decomposition into affine
spaces for every dimension vector $e$ and every indecomposable representation
$M$ of defect $-1$ and defect $0$, with exception of the non-Schurian
representations in homogeneous tubes. We characterize the affine spaces in
terms of the combinatorics of a fixed coefficient quiver for $M$. The method of
proof is to exhibit explicit equations for the Schubert cells of $Gr_e(M)$ and
to solve this system of equations successively in linear terms. This leads to
an intricate combinatorial problem, for whose solution we develop the theory of
Schubert systems.
  In the sequel to this paper, we extend the result of this paper to all
indecomposable representations $M$ of $Q$ and determine explicit formulae for
their $F$-polynomials.
