Weights, Kovalevskaya exponents and the Painlevé property

Weighted degrees of quasihomogeneous Hamiltonian functions the Painlevé equations are investigated. A t-uple positive integers, called a regular weight, satisfying certain conditions related to singularity theory is classified. For each polynomial equation weight associated. Conversely, for 2 and 4-dim cases, it shown that there exists differential property associated with weight. Kovalevskaya exponents systems also investigated by means weights, dynamical theory. It one-to-one correspondence between Laurent series solutions stable manifolds vector field obtained blow-up system. autonomous equations, level surface can be decomposed into disjoint union manifolds.