  We propose an Einstein-{\ae}ther scalar-tensor cosmological model. In
particular in the scalar-tensor Action Integral we introduce the {\ae}ther
field with {\ae}ther coefficients to be functions of the scalar field. This
cosmological model extends previous studies on Lorentz-violating theories. For
a spatially flat Friedmann--Lema\^{\i}tre--Robertson--Walker background space
we write the field equations which are of second-order with dynamical variables
the scale factor and the scalar field. The physical evolution of the field
equations depends upon three unknown functions which are related to the
scalar-tensor coupling function, the scalar field potential and the {\ae}ther
coefficient functions. We investigate the existence of analytic solutions for
the field equations and the integrability properties according to the existence
of linear in the momentum conservation laws. We define a new set of variables
in which the dynamical evolution depends only upon the scalar field potential.
Furthermore, the asymptotic behaviour and the cosmological history is
investigated where we find that the theory provides inflationary eras similar
with that of scalar-tensor theory but with Lorentz-violating terms provided by
the {\ae}ther field. Finally, in the new variables we found that the field
equations are integrable due to the existence of nonlocal conservation laws for
arbitrary functional forms of the three free functions.
