  Recently the most general completion of Brans-Dicke theory was appeared with
energy exchanged between the scalar field and ordinary matter, given that the
equation of motion for the scalar field keeps the simple wave form of
Brans-Dicke. This class of theories contain undetermined functions, but there
exist only three theories which are unambiguously determined from consistency.
Here, for the first such theory, it is found the action of the vacuum theory
which arises as the limit of the full matter theory. A symmetry transformation
of this vacuum action in the Jordan frame is found which consists of a
conformal transformation of the metric together with a redefinition of the
scalar field. Since the general family of vacuum theories is parametrized by an
arbitrary function of the scalar field, the action of this family is also
found. As for the full matter theory it is only found the action of the system
when the matter Lagrangian vanishes on-shell, as for example for pressureless
dust. Due to the interaction, this matter Lagrangian is non-minimally coupled
either in the Jordan or the Einstein frame.
