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