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[en] Within quantum-mechanical fission theory, P-odd and P-even correlations in angular distributions of products of the ternary fission of nuclei that is induced by polarized cold and thermal neutrons are investigated on the basis of a nonevaporative mechanism of third-particle emission and under the assumption that a two-humped fission barrier exists. It is shown that these correlations for third particles are induced by the analogous correlations for ternary-fission fragments, the latter being transferred to the third particle because of the kinematical conditions of third-particle emission that are associated with the charge and mass asymmetry of fragments. Optimum methods for observing the above correlations for third particles are discussed. The possibility of discovering the emission of prescission neutrons in the fission process against the background of evaporated neutrons by means of studying P-odd and P-even correlations is explored.