  In this paper we show that the $SL_{2}(\mathbb C)$ character variety of the
Riemann sphere $\Sigma_5$ with five boundary components is a $5$-parameter
family of affine varieties of dimension $4$. We endow this family of affine
varieties with an action of the braid group and classify exceptional finite
orbits. This action represents the nonlinear monodromy of the $2$ variable
Garnier system and finite orbits correspond to algebraic solutions.
