  Considering the general Lagrangian of k-essence models, we study and classify
them through variables connected to the fluid equation of state parameter
w_\kappa. This allows to find solutions around which the scalar field describes
a mixture of dark matter and cosmological constant-like dark energy, an example
being the purely kinetic model proposed by Scherrer. Making the stronger
assumption that the scalar field Lagrangian is exactly constant along solutions
of the equation of motion, we find a general class of k-essence models whose
classical trajectories directly describe a unified dark matter/dark energy
(cosmological constant) fluid. While the simplest case of a scalar field with
canonical kinetic term unavoidably leads to an effective sound speed c_s=1,
thereby inhibiting the growth of matter inhomogeneities, more general
non-canonical k-essence models allow for the possibility that c_s << 1 whenever
matter dominates.
