A kinetic model involving one order parameter was developed to study the impact of het- erogeneity on the nucleation. The role of asymmetry on phase transitions in the presence of an intermediate metastable state is described in comparison with the generic model. In particular, the discovery of anomalous generation and extinction of crystal nuclei in nonequilibrium supercooled liquids reflects experimentally the effect of the real heterogeneous structure and its irreversible relaxation on the stability of nuclei. It was found that the cubic term in the order parameter of the Landau-type potential associated with the asymmetry of the system serves to increase the stability of crystalline state for the suitably varied control parameter. The cases which allow analytical solution of the problem and the realization of different transition scenarios are also discussed.