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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 affinespaces for every dimension vector $e$ and every indecomposable representation$M$ of defect $-1$ and defect $0$, with exception of the non-Schurianrepresentations in homogeneous tubes. We characterize the affine spaces interms of the combinatorics of a fixed coefficient quiver for $M$. The method ofproof is to exhibit explicit equations for the Schubert cells of $Gr_e(M)$ andto solve this system of equations successively in linear terms. This leads toan intricate combinatorial problem, for whose solution we develop the theory ofSchubert systems.  In the sequel to this paper, we extend the result of this paper to allindecomposable representations $M$ of $Q$ and determine explicit formulae fortheir $F$-polynomials.

Quiver Grassmannians of type extended Dynkin type D - Part 1: Schubert systems and decompositions into affine spaces