  The Relativistic Motion Integrator (RMI) consists in integrating numerically
the EXACT relativistic equations of motion, for a given metric (corresponding
to a gravitational field at first post-Newtonian order or higher), instead of
Newtonian equations plus relativistic corrections. The aim of the present paper
is to validate the method, and to illustrate how RMI can be used for space
missions to produce relativistic ephemerides of test-bodies (or satellites).
Indeed, nowadays, relativistic effects have to be taken into account, and
comparing a RMI model with a classical keplerian one helps to quantify such
effects.
  LISA is a relevant example to use RMI. A precise orbit model for the LISA
spacecraft is needed not only for the sake of satellite ephemerides but also to
compute the photon flight time in laser links between spacecraft, required in
LISA data pre-processing in order to reach the gravitational wave detection
level.
  Relativistic effects in LISA orbit model needed to be considered and
quantified. Using RMI, we show that the numerical classical model for LISA
orbits in the gravitational field of a non-rotating spherical Sun without
planets can be wrong, with respect to the numerical relativisitic version of
the same model, by as much as about ten kilometers in radial distance during a
year and up to about 60 kilometers in along track distance after a year... with
consequences on estimated photon flight times.
  We validated RMI numerical results with a 1PN analytical developpement.
