The complex geometry of the Kowalewski-Painlevé analysis
An integrable system is algebraic complete integrable if its trajectories are straight line motions on complex algebraic tori. Improving a hundred years old result due to S. Kovalevskaya the authors establish conditions for algebraic complete integrability. The main theorem states that if the system \(z'=f(z)\) is algebraic complete integrable with invariant tori not containing elliptic curves then it has a ``coherent tree'' of Laurent solutions, and inversely, if a regular Hamiltonian system has the sufficient number of polynomial invariants in involution with a ``coherent tree'' of Laurent solutions then it is algebraic complete integrable. Conditions are given for the convergence of formal Laurent solutions. As an example a geodesic flow on SO4 is treated in detail.
scientific article; zbMATH DE number 4111637