of this introduction we fix some notations and outline the content of the sections to follow. Let us briefly comment on our choice of notation and style of presentation. Our original motivation for studying parameter-dependent linear ODEs derives from our desire to develop a more rigorous theory of Darboux transformations for (1+1)-dimensional linear evolution equations than the existing ones; cf. . This explains why we primarily focus on scalar equations and just outline the corresponding results for systems (contrary to the standard approach in the ODE literature). It also justifies the notation of variables, $x$ for the independent variable and $t$ for the parameter, as well as their order.

Given a subset $U$ of the $(t,x)$-plane, the projection $U$ to the $t$-axis is denoted by ${\mathop{\rm pr}\nolimits}_tU$, where ${\mathop{\rm pr}\nolimits}_t\colon\mathbb R^2\to\mathbb R$ is the projection $(t,x)\mapsto t$, and for each $t_0\in{\mathop{\rm pr}\nolimits}_tU$ the set $U_{t_0}$ is the projection of the section of $U$ by the line $t=t_0$ to the $x$-axis, $$\begin{gathered}
{\mathop{\rm pr}\nolimits}_tU:=\{t\in\mathbb R\mid \exists\ x\in\mathbb R\colon (t,x)\in U\},\quad
{\mathop{\rm pr}\nolimits}_tU\ni t\mapsto U_t:=\{x\in\mathbb R\mid (t,x)\in U\}.\end{gathered}$$ For a function $g\colon U\to\mathbb R$, the expression “$g\ne0$ on $U$” means that $g(z)\ne0$ for any $z\in U$.

For an open set $\Omega$ of 