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We reduce a question of Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich,which asks for a classification of all $\text{SL}_2(\mathbb{R})$-invariantergodic probability measures with completely degenerate Kontsevich-Zorichspectrum, to a conjecture of M\"oller's. Let $\mathcal{D}_g (1)$ be the subsetof the moduli space of Abelian differentials $\mathcal{M}_g$ whose elementshave period matrix derivative of rank one. There is an$\text{SL}_2(\mathbb{R})$-invariant ergodic probability measure $\nu$ withcompletely degenerate Kontsevich-Zorich spectrum, i.e. $\lambda_1 = 1 >\lambda_2 = \cdots = \lambda_g = 0$, if and only if $\nu$ has support containedin $\mathcal{D}_g (1)$. We approach this problem by studying Teichm\"ullerdiscs contained in $\mathcal{D}_g (1)$. We show that if $(X,\omega)$ generatesa Teichm\"uller disc in $\mathcal{D}_g (1)$, then $(X,\omega)$ is completelyperiodic. Furthermore, we show that there are no Teichm\"uller discs in$\mathcal{D}_g (1)$, for $g = 2$, and the two known examples of Teichm\"ullerdiscs in $\mathcal{D}_g (1)$, for $g = 3, 4$, are the only two such discs inthose genera. Finally, we prove that if there are no genus five Veech surfacesgenerating Teichm\"uller discs in $\mathcal{D}_5(1)$, then there are noTeichm\"uller discs in $\mathcal{D}_g (1)$, for $g = 5,6$.

Teichm\"uller Discs with Completely Degenerate Kontsevich-Zorich Spectrum