In the context of generalised Brans-Dicke cosmology we use the Killing tensors of the minisuperspace in order to determine the unspecified potential of a scalar-tensor gravity theory. Specifically, based on the existence of contact symmetries of the field equations, we find four types of potentials which provide exactly integrable dynamical systems. We investigate the dynamical properties of these potentials by using a critical point analysis and we find solutions which lead to cosmic acceleration and under specific conditions we can have de-Sitter points as stable late-time attractors.Comment: 15 pages; 2 figures; accepted for publication by Phys. Rev.

In the context of Einstein-aether scalar field cosmology we solve the field equations and determine exact and analytic solutions. In particular, we consider a model proposed by Kanno and Soda where the aether and the scalar fields interact through the aether coefficient parameters, which are promoted to be functions of the scalar field. For this model, we write the field equations by using the minisuperspace approach and we determine the scalar field potentials which leads to Liouvilleintegrable systems. We solve the field equations for five families of scalar field potentials and, whether it is feasible, we write down the analytic solutions by using closed-form functions. 95.35.+d, 95.36.+x

1 The Hubble parameter in the extended Finsler-Randers model is given by H 2 (a) = 8πG 3 ρm − a −2 aΨ(a)da − C 1 a 2 [see Eqs.(32) and (39) in Ref. [14]].

We study the dynamical properties of a large body of varying vacuum cosmologies for which dark matter interacts with vacuum. In particular, performing the critical point analysis we investigate the existence and the stability of cosmological solutions which describe de-Sitter, radiation and matter dominated eras. We find several cases of varying vacuum models that admit stable critical points, hence they can be used in describing the cosmic history. 95.35.+d, 95.36.+x

We perform for the first time a dynamical system analysis of both the background and perturbation equations, of ΛCDM cosmology and quintessence scenario with an exponential potential. In the former case the perturbations do not change the stability of the late-time attractor of the background equations, and the system still results in the dark-energy dominated, de Sitter solution, having passed from the correct dark-matter era with γ ≈ 6/11. However, in the case of quintessence the incorporation of perturbations changes the stability and properties of the background evolution, and the only conditionally stable points present either an exponentially increasing matter clustering not favored by observations, or Laplacian instabilities, and thus not physically interesting. This result is a severe disadvantage of quintessence cosmology comparing to ΛCDM paradigm.