  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.
