But this contradicts the fact that $\psi(t,\theta(t))=1$ for $t\in{\mathop{\rm pr}\nolimits}_tU_1\cap{\mathop{\rm pr}\nolimits}_tU_2$. Therefore, for any $P\in{{\rm DO}}(\Omega)$ with such an $\Omega$, the equation $Pu=0$ possesses no (global) fundamental set of solutions on $\Omega$.

Henceforth we may therefore assume that some connected component of $\Omega$ is not an $x$-simple region. Applying Lemma \[lem:OnOpenConnectedNonX-SimpleSets\] to this component, we get that up to reflections in $t$, the set $\Omega$ contains, for some $\tilde t_0,\varepsilon,\tilde x_1,\tilde x_2\in\mathbb R$ with $\varepsilon>0$, $\tilde x_1\leqslant\tilde x_2$ and for some closed subset $\Upsilon$ of the line segment $\{\tilde t_0\}\times[\tilde x_1,\tilde x_2]$ with $(\tilde t_0,\tilde x_1),(\tilde t_0,\tilde x_2)\in\Upsilon$, the subset $[\tilde t_0-\varepsilon,\tilde t_0]\times[\tilde x_1-\varepsilon,\tilde x_2+\varepsilon]\setminus\Upsilon$ and does not intersect $\Omega$. Consider any $P\in{{\rm DO}}(\Omega)$ with $G(t)\to-\infty$ as $t\to\tilde t_0{}^-$,[^3] where $$G(t):=\int_{\tilde x_1-\varepsilon}^{\tilde x_2+\varepsilon}\frac{g^{p-1}(t,x)}{g^p(t,x)}\,{\rm d}x,
\quad t\in[\tilde t_0-\varepsilon,\tilde t_0),$$ $p={\mathop{\rm ord}\nolimits}P$, and $g^p$ and $g^{p-1}$ denote the leading and subleading coefficients of $P$, respectively. An example of such coefficients is given by $g^p(t,x)=1$ and $g^{p-1}(t,x)=-c\big((x-\tilde x_1)^2+(t-\tilde t_0)^2\big)^{-1}$ with $c>0$ for $(t,x)\in\Omega$, cf. Example \[ex:1stOrderHomLinODEWithVanishingWronskian\]. Then $\mathrm W(\varphi^1,\dots,\varphi^p)=0$ on $\{\tilde t_0\}\times[\tilde x_1-\varepsilon,\tilde x_1)$ for any solutions $\varphi^1$, …, $\varphi^p$ of the equation $Pu=0$. Indeed, it suffices to prove this claim only for the point $(\tilde t_0,\tilde x_1-\varepsilon)$. Supposing that it is not 