  The geometry of the Ellis-Bronnikov wormhole is implemented in the Rastall
and k-essence theories of gravity with a self-interacting scalar field. The
form of the scalar field potential is determined in both cases. A stability
analysis with respect to spherically symmetric time-dependent perturbations is
carried out, and it shows that in k-essence theory the wormhole is unstable,
like the original version of this geometry supported by a massless phantom
scalar field in general relativity. In Rastall's theory, it turns out that a
perturbative approach reveals the same inconsistency that was found previously
for black hole solutions: time-dependent perturbations of the static
configuration prove to be excluded by the equations of motion, and the wormhole
is, in this sense, stable under spherical perturbations.
