  The k-essence theory with a power-law function of $(\partial\phi)^2$ and
Rastall's non-conservative theory of gravity with a scalar field are shown to
have the same solutions for the metric under the assumption that both the
metric and the scalar fields depend on a single coordinate. This equivalence
(called k-R duality) holds for static configurations with various symmetries
(spherical, plane, cylindrical, etc.) and all homogeneous cosmologies. In the
presence of matter, Rastall's theory requires additional assumptions on how the
stress-energy tensor non-conservation is distributed between different
contributions. Two versions of such non-conservation are considered in the case
of isotropic spatially flat cosmological models with a perfect fluid: one (R1)
in which there is no coupling between the scalar field and the fluid, and
another (R2) in which the fluid separately obeys the usual conservation law. In
version R1 it is shown that k-R duality holds not only for the cosmological
models themselves but also for their adiabatic perturbations. In version R2,
among other results, a particular model is singled out that reproduces the same
cosmological expansion history as the standard $\Lambda$CDM model but predicts
different behaviors of small fluctuations in the k-essence and Rastall
frameworks.
