The paper separates a class of finite-dimensional systems of nonlinear differential equations, the exact analytical solution of which can be represented in the form of quadratures. The paper uses a particular case of the system of the separated class as a set of equality constraints for the problem of optimal management of a closed finite-dimensional labor market with a common selection coefficient - the management parameter for the system under study. The paper specifies the definitions of the qualification categories for labor market subjects, with allowance for the physical meaning of their behavior in the system under study. It introduces quality factors for meeting the demand for labor, which are the averaged difference between the remuneration of labor and proceeds of the activities of subjects at each of the three qualification categories. It introduces a quality function in respect of the management of the labor market system, which is a sum of the products of the functions of the shares owned by the subjects at each of the three qualification categories by their quality coefficients. It considers labor markets with different ratios of quality factors. The case where the quality factor of subjects of a low qualification category is higher than that of a high qualification category has been shown to contradict the physical meaning of the model. Quality function vs management parameter curves are plotted for each labor market system under study. The paper gives examples of real-life labor markets for every physically admissible ratio of quality factors. The optimal management of the labor market system is shown to not necessarily imply that the management parameter tends to its extreme values. The paper plots a management quality function for a real-life labor market with a city-forming enterprise exemplified by the labor market of the village of Sylva, the Perm Territory, and determines the optimal values of management parameters.

systems of nonlinear differential equations, integrability by quadratures, labor market model, optimal management problem, price – quality ratio, management efficiency, staff dynamics, forecasting the labor market, selection coefficient, quality coefficient.