  We investigate the existence of anisotropic self-similar exact solutions in
symmetric teleparallel $f\left( Q\right)$-theory. For the background geometry
we consider the Kantowski-Sachs and the Locally Rotationally Symmetric Bianchi
type III geometries. These two anisotropic spacetimes are of special interest
because in the limit of isotropy they are related to the closed and open
Friedmann--Lema\^{\i}tre--Robertson--Walker cosmologies respectively. For each
spacetime there exist two distinct families of flat, symmetric connections,
which share the symmetries of the spacetime. We present the field equations,
and from them, we determine the functional form of the $f\left( Q\right)$
Lagrangian which yields self-similar solutions. We initially consider the
vacuum case and subsequently we introduce a matter source in terms of a perfect
fluid. Last but not least, we report some self-similar solutions corresponding
to static spherically symmetric spacetimes.
