  In this paper we show that, for general scalar fields, stationary
configurations are possible for shift symmetric theories only. This symmetry
with respect to constant translations in field space should either be manifest
in the original field variables or reveal itself after an appropriate field
redefinition. In particular this result implies that neither k-Essence nor
Quintessence can have exact steady state / Bondi accretion onto Black Holes. We
also discuss the role of field redefinitions in k-Essence theories. Here we
study the transformation properties of observables and other variables in
k-Essence and emphasize which of them are covariant under field redefinitions.
Finally we find that stationary field configurations are necessarily linear in
Killing time, provided that shift symmetry is realized in terms of these field
variables.
