  A scalar theory of gravitation with a preferred reference frame (PRF) is
considered, that accounts for special relativity and reduces to it if the
gravitational field cancels. The gravitating system consists of a finite number
of perfect-fluid bodies. An " asymptotic " post-Newtonian (PN) approximation
scheme is used, allowing an explicit weak-field limit with all fields expanded.
Exact mass centers are defined and their exact equations of motion are derived.
The PN expansion of these equations is obtained: the zero-order equations are
those of Newtonian gravity (NG), and the equations for the first-order (PN)
corrections depend linearly on the PN fields. For PN corrections to the motion
of the mass centers, especially in the solar system, one may assume "
very-well-separated " rigidly moving bodies with spherical self-fields of the
zero-order approximation. The PN corrections reduce then to a time integration
and include spin effects, which might be significant. It is shown that the
Newtonian masses are not correct zero-order masses for the PN calculations. An
algorithm is proposed, in order to minimize the residual and to assess the
velocity in the PRF.
