Anisotropic spacetimes in $f(T,B)$ theory IV: Noether symmetry analysis

The Noether symmetry analysis is applied for the analysis of the field equations in an anisotropic background in $f(T,B)$-theory. We consider the $f\left( T,B\right) =T+F\left( B\right) $ which describes a small deviation from TEGR introduced by the boundary scalar $B$. For the Bianchi\ I, Bianchi III and Kantowski-Sachs geometries there exists a minisuperspace description and Noether's theorems are applied. We investigate the existence of invariant point transformations. We find that for the Bianchi I spacetime the gravitational field equations are Liouville integrable for the $F\left( B\right) =-\frac{B}{\lambda}\ln B$ theory. The analytic solution is derived and the application of Noether symmetries to the Wheeler-DeWitt equation of quantum cosmology is discussed.