On the weak Kowalevski-Painleve property for hyperelliptically separable systems

We consider so called hyperelliptically separable systems (h.s.s.) arising in various physical problems, whose generic invariant manifolds can be completed either to hyperelliptic Jacobians or to their nonlinear subvarieties (strata) or their finite coverings. In the case of strata the algebraic geometrical structure of such systems has much in common with that of algebraic completely integrable systems (a.c.i.s.). Using this property we study formal singular solutions of a.c.i.s. and h.s.s., which may contain fractional powers of time. We give estimates for the number and leading behavior of their principal and lower balances both for a generic and for the so called physical direction of the flow. This can be regarded as an useful extension of the Kowalevski-Painleve integrability test. We also prove that when the system is h.s. but not a.c.i., its generic solutions are single-valued on an infinitely sheeted ramified covering of the complex time plane. Some model examples are considered, such as the hierarchy of integrable generalizations of the Henon-Heiles and the Neumann systems.