Abstract <p>The study of the existence of the quadratic local integral in stationary two-dimensional potential fields that was initiated in the first part of the work, is continued. New mathematical relationships that deepen the understanding of the structure of functions describing the behavior of potential fields under arbitrary mass distribution have been proposed. The rotation of the coordinate system to simplify the equations and emphasize key features of the functional dependencies has been employed. Particular attention has been paid to arbitrary functions defining the potential and its derivatives under specific conditions. Their properties and possible solutions have been analyzed. Besides, linear differential equations with polynomial and periodic solutions have been studied. Theoretical results, which can be used for further analysis of quadratic integrals and for clarifying the differences between polynomials and other types of functions in broader mathematical models, have been formulated. The paper is partially based on a report presented at the Modern Stellar Astronomy 2024 conference.</p>