International audienceWe present the radial-velocity follow-up of two Kepler planetary transiting candidates (KOI-189 and KOI-686) carried out with the SOPHIE spectrograph at the Observatoire de Haute Provence. These data promptly discard these objects as viable planet candidates and show that the transiting objects are in the regime of very low-mass stars, where a strong discrepancy between observations and models persists for the mass and radius parameters. By combining the SOPHIE spectra with the Kepler light curve and photometric measurements found in the literature, we obtain a full characterization of the transiting companions, their orbits, and their host stars. The two companions are in significantly eccentric orbits with relatively...