The Third, Fifth and Sixth Painlevé Equations on Weighted Projective Spaces

The third, fifth and sixth Painlevé equations are studied by means of the weighted projective spaces $\mathbb C P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton polyhedrons of the equations. Singular normal forms of the equations, symplectic atlases of the spaces of initial conditions, Riccati solutions and Boutroux's coordinates are systematically studied in a unified way with the aid of the orbifold structure of $\mathbb C P^3(p,q,r,s)$ and dynamical systems theory.