Anisotropic Solutions in Symmetric Teleparallel $f\left( Q\right)$-theory: Kantowski-Sachs and Bianchi III LRS Cosmologies

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.