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A $5/4$ commensurability of KIC 5773205, the smallest eclipsing red dwarf detected by the Kepler mission

KIC 5773205 is the least luminous eclipsing M dwarf found in the Villanova catalog of eclipsing binaries detected by the {\it Kepler} mission. We processed and analyzed the three available quarters of mission data for this star and discovered a persistent periodic variation of the light curve with a period, which is in exact 4:5 commensurability to the orbital period. Three routes of interpretation are considered: 1) non-radial pulsations excited by the tidal interaction at a specific eigenfrequency; 2) a high-order spin-orbit resonance caused by the tides; 3) an ellipsoidal deformation caused by an outer orbiting companion in a mean motion resonance. All three explanations meet considerable difficulties, but the available facts seem to favor the tidally driven pulsation scenario. The star may represent a new type of heartbeat binary with tidally excited pulsations that are close to the orbital motion in frequency.