  Given that the simple wave equation of Brans-Dicke theory for the scalar
field is preserved, we have investigated, through exhaustively analyzing the
Bianchi identities, the consistent theories which violate the exact energy
conservation equation. It is found that only three theories exist which are
unambiguously determined from consistency, without imposing arbitrary functions
by hand. Each of these theories possesses a specific interaction term which
controls the energy exchange between the scalar field and ordinary matter. The
theories contain new parameters (integration constants from the integration
procedure) and when these are switched-off, Brans-Dicke theory emerges. As
usually, the vacuum theories can be defined from the complete Brans-Dicke
theories when the matter energy-momentum tensor vanishes.
