  In a higher-order modified teleparallel theory cosmological we present
analytical cosmological solutions. In particular we determine forms of the
unknown potential which drives the scalar field such that the field equations
form a Liouville integrable system. For the determination of the conservation
laws we apply the Cartan symmetries. Furthermore, inspired from our solutions,
a toy model is studied and it is shown that it can describe the Supernova data,
while at the same time introduces dark matter components in the Hubble
function. When the extra matter source is a stiff fluid then we show how
analytical solutions for Bianchi I universes can be constructed from our
analysis. Finally, we perform a global dynamical analysis of the field
equations by using variables different from that of the Hubble-normalization.
