  We consider a Lorentz violating scalar field cosmological model given by the
modified Einstein-aether theory defined in Weyl integrable geometry. The
existence of exact and analytic solutions is investigated for the case of a
spatially flat Friedmann--Lema\^{\i}tre--Robertson--Walker background space. We
show that the theory admits cosmological solutions of special interests. In
addition, we prove that the cosmological field equations admit the Lewis
invariant as a second conservation law, which indicates the integrability of
the field equations.
