differential equations for which not all initial conditions $x(t_0)=x_0$ are possible: there is a proper subset $C\subset (-\infty, +\infty)\times X$ of allowed initial conditions. This is the case, i.e., for the Clairaut equations. We discuss a particular, illustrative example in the next section. However, it is not hard to see that Theorems \[thm:main\] and \[thm:2\] can be adapted for such situations.

We can introduce the local process maps $U(t_1, t_0)$ as in the previous section: $U(t_1, t_0)(a) = {\frak u}(t_1; t_0, a)$. However, now $U(t_1, t_0)$ may not be defined on all of $X$ and, for every $a$, the range of admissible $t_1$ will be different. If we restrict $t_0$ to a compact set $[\alpha, \beta]\subset [0, +\infty)$ and $a$ to a compact set $K\subset X$, then, thanks to assumption [**TT**]{}, the infimum of $ T(t_0, a)$ over $[\alpha, \beta]\times K$ is strictly positive, and this gives a non-trivial admissible interval for $t_1$ when $(t_0, a)\in [\alpha, \beta]\times K$. At the moment we do not see any benefits in constructing an abstract theory of such local (semi)-processes.

Example {#sec:ex}
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The Clairaut equation is a scalar ODE of the form \[eq:cl-gen\] x = t[x]{} + ([x]{}) with some function $\psi$. 