Second Degree Local Integral for Rotating Systems. Part I

The results of a study on the existence of a local integral quadratic relative to velocity components for stationary stellar systems in the absence of any symmetry of the gravitational potential are presented. This refers to a separate isolated invariant surface in phase space, allowing the velocity field to be determined immediately for certain specific initial conditions. All two-dimensional potentials that allow such a local integral for rotating systems are listed. In this first publication, we will limit ourselves to cases of constant energy expansion in powers and coinciding characteristics, which lead to both known and non-trivial potentials allowing a quadratic local integral. The work is partially based on a presentation at the Modern Stellar Astronomy 2024 conference.