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A huge family of separable potential perturbations of integrable billiard systems and the Jacobi problem for geodesics on an ellipsoid is given through the Appell hypergeometric functions F4 of two variables, leading to an interesting connection between two classical theories: separability in Hamilton–Jacobi sense on the one hand and the theory of hypergeometric functions on the other.

Separabilty in Hamilton­Jacobi Sense in Two Degrees of Freedom and the Appell Hypergeometric Functions

Separabilty in HamiltonJacobi Sense in Two Degrees of Freedom and the Appell Hypergeometric Functions