Aims, We examine a recently-proposed evolutionary scenario that could explain the existence of the low-mass, helium-enriched PG 1159 stars. We focus in particular on studying the pulsational stability properties of the evolutionary models predicted by such a scenario. Methods. We assess the overstability of pulsation g-modes of stellar models as evolution proceeds in the PG 1159 domain. Stellar models are extracted from the full evolution of a 1-M⊙ model star that experiences its first thermal pulse as a late thermal pulse (LTP) after leaving the AGB. The evolutionary stages corresponding to the born-again episode and the subsequent helium sub-flashes are taken into account in detail. Results. Under reasonable assumptions of mass-loss rate, the evolutionary scenario reproduces the high helium abundances observed in some PG 1159 stars. We find that, despite the high helium abundance in the driving layers, a narrow region exists in the log T<SUB>eff</SUB> - log g diagram for which the helium-enriched PG 1159 sequence exhibits unstable pulsation modes with periods in the range 500 to 1600 s. In particular, the nonpulsating helium-enriched PG 1159 star, MCT 0130-1937, is located outside the theoretical instability domain. Our results suggest that MCT 0130-1937 is a real non-pulsating star and that the lack of pulsations should not be attributed to unfavorable geometry. Conclusions. Our study hints at a consistent picture between the evolutionary scenario that could explain the existence of heliumenriched PG 1159 stars and the nonvariable nature of MCT 0130-1937. We also present theoretical support for the unusually high helium abundance observed in the nonpulsating PG 1159 star HS 1517+7403. We suggest that HS 1517+7403 could be a transition object linking the low-mass helium-rich O(He) stars with the helium-enriched PG 1159 stars via the evolutionary connection K1-27 →HS 1517+7403 → MCT 0130-1937.Facultad de Ciencias Astronómicas y GeofísicasInstituto de Astrofísica de La Plata