Dynamics and exact Bianchi I spacetimes in Einstein-æther scalar field theory

We determine exact and analytic solutions of the gravitational field equations in Einstein-aether scalar model field with a Bianchi I background space. In particular, we consider nonlinear interactions of the scalar field with the aether field. For the model under consideration we can write the field equations by using the minisuperspace description. The point-like Lagrangian of the field equations depends on three unknown functions. We derive conservation laws for the field equations for specific forms of the unknown functions such that the field equations are Liouville integrable. Furthermore, we study the evolution of the field equations and the evolution of the anisotropies by determining the equilibrium points and analyzing their stability.