  We analyze the implications of having a divergent speed of sound in k-essence
cosmological models. We first study a known theory of that kind, for which the
Lagrangian density depends linearly on the time derivative of the k-field. We
show that when k-essence is the only source consistency requires that the
potential of the k-field be of the inverse square form. Then, we review the
known result that the corresponding power-law solutions can be mapped to
power-law solutions of theories with no divergence in the speed of sound. After
that, we argue that the requirement of a divergent sound speed at some point
fixes uniquely the form of the Lagrangian to be exactly the one considered
earlier and prove the asymptotic stability of the most interesting solutions
belonging to the divergent theory. Then, we discuss the implications having not
just k-essence but also matter. This is interesting because introducing another
component breaks the rigidity of the theory, and the form of the potential
ceases to be unique as happened in the pure k-essence case. Finally, we show
the finiteness of the effective sound speed under an appropiate definition.
