In this paper we show that the SL2pCq character variety of the Riemann sphere Σ5 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 its algebraic solutions.
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