  Unimodular gravity became an object of increasing interest in the late
$80$-ties and was recently used in primordial Universe modeling with
cosmological constant, in the context of the Brans-Dicke gravity including
scalar field. In the present article we investigate the possibility of imposing
the unimodular condition within the $5$-dimensional Kaluza-Klein theory
including the scalar field. The variational principle is formulated in $5$
dimensions first, and dimensional reduction is applied to the resulting set of
equations. A cosmological model based on these equations is then presented and
discussed.
