Symmetry and the Singular Point Analysis for Ordinary Differential Equations

We show a direct relation between the singular point analysis and the symmetry for ordinary differential equations. It is proved that a system with a meromorphic solution allows Laurent solutions depending on m arbitrary parameters if it has m independent symmetries, regardless of the existence of single-valued first integrals. Applying the result to Hamiltonian systems, we show a pairing property and introduce a new definition of integrability as a synthesis of symmetries and first integrals. Further we deal with integrable Hamiltonian systems whose first integrals are in involution and discuss a relation between their symmetries and infinitely multi-valued first integrals yielded by Liouville's quadrature.