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.
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