Rational integrals of 2-dimensional geodesic flows: new examples

The paper reduces {F,H}=0 to a solvable PDE for ψ and then constructs ψ via Bessel-function ansätze, yielding Λ=ψ_x^2+ψ_y^2 and the rational integral F with coefficients f1,f2,g1,g2 as in equations (2.15)–(2.16) and (4.3) . The candidate solution instead proves directly in isothermal coordinates that two linear-in-momenta quantities L and M share the same Poisson multiplier σ whenever ψ satisfies the transformed PDE ϕ_{uu}+ϕ_{vv}+(1/v)ϕ_v=0 (the paper’s (3.1)), hence M/L is conserved; it then identifies L and M with the paper’s numerator/denominator (up to the harmless common scale) and matches the analyticity claims . The arguments agree on hypotheses and conclusions; the proofs are different but consistent.