Properties of Mn 3XN and Mn 3XC types of compounds are analyzed on the basis of a model for the electronic band structure, which consists of a large conduction band overlapping a narrow band that results from the strong hybridization between the p orbitals of the metalloid and some of the d orbitals of the manganese. The crystal field at the sites of manganese is assumed to be strong. The structure of the narrow band is calculated in the tight-binding model. The Fermi energy lies very close to a sharp singularity in the density of states. In the cubic and Pauli paramagnetic phase, such a singularity has a sixfold degeneracy. The magnetic and structural instabilities, which appear when the temperature is decreased, are explained by the removal of that degeneracy by a shear strain and the formation of small magnetic moments. The phase transitions can be studied in detail by expanding the variation of the free energy with respect to the shear strains and the magnetic moments. The coefficients of the expansion are calculated as functions of the temperature. The variation of the volume is explained by the existence of coupling terms to the shear strain and to the magnetic moments.