### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 019, 22 pages      arXiv:1506.00444      https://doi.org/10.3842/SIGMA.2016.019

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

Hayato Chiba
Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan

Received September 17, 2015, in final form February 18, 2016; Published online February 23, 2016

Abstract
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.

Key words: Painlevé equations; weighted projective space.

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