We consider the problems of consistency of the results of international comparisons, equivalence of the national standards, and improvement of the methods aimed at the realization of the main fixed points of solidification of the metals on the International Temperature Scale (ITS-90). We identified two sources of uncertainties of temperature values in the realization of the fixed points of solidification of the metals. We carried out experiments aimed at the initiation of the inner interface between the liquid and solid phases of the metal and studied the influence of the conditions of initiation on the structure of the interface and the measured values of temperature in the aluminum and indium fixed-point cells. The problem of correctness of evaluation of the influence of impurities according to the regularities established for a single outer interface of crystallization moving from the walls of the crucible toward its center is investigated. The analysis of solidification curves obtained for different initiation conditions makes it possible to establish a criterion for the existence of a continuous inner interface and the conditions of formation solely of the outer interface (without formation of the inner interface) in the course of slow crystallization of the metal. The obtained criteria can be used by the metrological institutes to choose correct conditions for the initiation of the inner interface in the case of application of cells and heaters of different designs.