  We investigate anisotropic and homogeneous cosmological models in the
scalar-tensor theory of gravity with non-minimal kinetic coupling of a scalar
field to the curvature given by the function $\eta\cdot(\phi/2)\cdot
G_{\mu\nu}\,\nabla^\mu \nabla^\nu \phi$ in the Lagrangian. We assume that the
space-times are filled a global unidirectional electromagnetic field that
minimally interacts with the scalar field. The Horndeski theory allows
anisotropy to grow over time. The question arises about isotropization. In the
theory under consideration, a zero scalar charge imposes a condition on the
anisotropy level, namely its dynamics develops in a limited region. This
condition uniquely determines a viable branch of solutions of the field
equations. The magnetic energy density that corresponds to this branch is a
bounded function of time. The sign of parameter $l=1+8\pi \varepsilon\eta
\Lambda$ determines the properties of cosmological models, where $\Lambda$ is
the cosmological constant. The sign $\varepsilon=\pm 1$ defines the canonical
scalar field and the phantom field, respectively. An inequality
$\varepsilon/\eta>0$ is a necessary condition for isotropization of models, but
not sufficient. Three cases were considered: $l=0$, the isotropization does not
occur; $l>0$, there is isotropization; $l<0$, there are two branches, one of
which has the property of isotropization, and the other describes the Universe
with collapse. The model with $l<0$ has an anisotropic bounce and the Universe
evolution begins with a non-zero volume value. Models of an infinitely
expanding Universe eventually go into inflation mode with the scale factor
$a(t)\propto e^{h_\eta\cdot t}$, $h_\eta=\sqrt{\frac{\varepsilon}{3\eta}}$.
