Finite orbits of the pure braid group on the monodromy of the 2-variable Garnier system

In this article, we realize the SL2(C) character variety of the Riemann sphere with five boundary components as a 5-parameter family of affine varieties of dimension
4. We show that the action of the mapping class group corresponds to certain action of the braid group on this family of affine varieties and classify exceptional finite orbits. This action represents the nonlinear monodromy of the 2-variable Garnier system and finite orbits correspond to its algebraic solutions.