  Every finite branch solutions to the sixth Painleve equation around a fixed
singular point is an algebraic branch solution. In particular a global solution
is an algebraic solution if and only if it is finitely many-valued globally.
The proof of this result relies on algebraic geometry of Painleve VI,
Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces,
resolutions of Kleinian singularities, and power geometry of algebraic
differential equations. In the course of the proof we are also able to classify
all finite branch solutions up to Backlund transformations.
