i.e., view solitons as continuous (infinite, one-sided) paths in $X$. If there is no uniqueness, the solutions/integral curves starting at the point $a$ form an integral funnel, $S(a)$, a subset of all paths starting at $a$. (Analysis of integral funnels for ODEs was initiated by H. Kneser in the 1920s, [@Kneser].) Denote by $\Omega$ the space of all continuous (infinite, one-sided) paths in $X$. Then the map $a\mapsto S(a)$ is a set-valued map (other names: multifunction, correspondence) from $X$ into $2^\Omega$. It has the (already mentioned) properties: if the path $w$ is in $S(a)$, then its shift, $\theta_\tau w$ is in $S(w(\tau))$, and if $u\in S(a)$ and $v\in S(u(t_1))$, then $u \underset{t_1}{\bowtie} v\in S(a)$.

An interesting and important question is whether it is possible to select a solution $u(\cdot, a)$ from every funnel $S(a)$ in such a way, that $u(t, a)$ has the semigroup property . In other words, is it possible to define a semigroup (semiflow) $U(t)$ so that, for every $a\in X$, $u(t, a) = U(t)(a)$, $t\ge 0$, is a path in $S(a)$? In [@C; @C-K], we show that the answer is yes under some very general assumptions. Moreover, we show that the selection $a\mapsto u(\cdot, a)\in S(a)$ 