The paper deals with the dynamics of conservative-degree-of-freedom vibro-impact systems involving one unilateral contact condition and a linear free flow. Among all possible trajectories, grazing linear modes (GLM) exhibit a contact occurrence with a vanishing incoming velocity which generates mathematical difficulties. We show that the First Return Time (FRT) to the chosen Poincaré section of the orbits which are sufficiently close to a GLM is always near a multiple of the fundamental period of the mode. This yields an infinite partition of the neighborhood of the GLMs on the Poincaré section along with square root singularities of the FRT depending on each set of the partition.