Monodromy of certain Painleve' VI transcendents and reflection groups

We study the global analytic properties of the solutions of a particular family of Painleve’ VI equations with the parameters $\beta =\gamma =0$, $\delta =\frac{1}{2}$ and $\alpha$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painleve’ VI equation.