Let $S$ be the dyadic solenoid.
Let $x\in S$, and let $X$ be the union of all arcs (homeomorphic copies of $[0,1]$) in $S$ containing $x$.
$X$ is called a composant of $S$.
It is well-known that $X$ is a dense first category one-to-one continuous image of the reals, and that $S$ is a homogeneous continuum.
Now let $y$ and $z$ be any two points in $S\setminus X$.
Is there a self-homeomorphism $h:S\to S$ such that $h[X]=X$ and $h(y)=z$?