The Cauchy-Lorentz family is the unique connected two-dimensional family of continuous probability densities invariant under projective transport induced by Riccati dynamics, via reformulation on the circle and stereographic projection.
Hille, Ordinary Differential Equations in the Com- plex Domain, A Wiley-Interscience publication (Wiley- Interscience, New York, 1976) see Chapter 4 for Riccati equations
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Geometric Origin of Exact Mean-Field Reductions: M{\"o}bius Symmetry and the Lorentzian Ansatz
The Cauchy-Lorentz family is the unique connected two-dimensional family of continuous probability densities invariant under projective transport induced by Riccati dynamics, via reformulation on the circle and stereographic projection.