  We study spatially flat isotropic universes driven by k-essence. It is shown
that Friedmann and k-field equations may be analytically integrated for
arbitrary k-field potentials during evolution with a constant baryotropic
index. It follows that there is an infinite number of dynamically different
k-theories with equivalent kinematics of the gravitational field. We show that
there is a large "window" of stable solutions, and that the dust-like behaviour
separates stable from unstable expansion. Restricting to the family of power
law solutions, it is argued that the linear scalar field model, with constant
function F, is isomorphic to a model with divergent speed of sound and this
makes them less suitable for cosmological modeling than the non-linear k-field
solutions we find in this paper.
