On the connection between hyperelliptic separability and Painleve integrability

We consider systems of ODEs which are associated with some physically signi ficant examples: shallow water equilibrium solutions, travelling waves of t he Harry Dym equation, a Lotka-Volterra system of competing species and the geodesic flow on the triaxial ellipsoid. The first three are shown to shar e the following properties: (i) they are hyperelliptically separable system s (HSS) and, after a suitable nonlinear time transformation, become algebra ically completely integrable (ACI) and (ii) they are of the weak Painleve t ype and become full Painleve after the application of this transformation. The geodesic flow on the other hand, although it passes the usual Painleve test, does not possess a full set of free constants and thus one may not co nclude whether it has the Painleve property or not. This system is also HSS and becomes ACI after the application of a suitable nonlinear time transfo rmation. We also combine our geometric-analytical investigation with a nume rical analysis of the system in the complex plane and show that there is pe rfect correspondence between the results of the two approaches. This corres pondence strengthens the reliability of such numerical studies and helps us better understand their implication in cases where such nonlinear transfor mations to complete integrability are not available.