This work continues the investigation of existence conditions for local integrals of motion quadratic in velocities within two-dimensional potential, with particular emphasis on rotating gravitational systems, as initiated in 1. We analyze the algebraic and analytic structure of the functions defining the system, along with the differential equations governing the potential dynamics. Special attention is paid to the polynomial nature, analyticity of solutions, and asymptotic behavior of these functions.
F. T. Shamshiev (Mon,) studied this question.