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.
Letac, Which functions preserve Cauchy laws?, Pro- ceedings of the American Mathematical Society67, 277 (1977)
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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.