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Let X be a 2-sphere with n punctures. We classify all conjugacy classes ofZariski-dense representations $$\rho: \pi_1(X)\to SL_2(\mathbb{C})$$ withfinite orbit under the mapping class group of X, such that the local monodromyat one or more punctures has infinite order. We show that all suchrepresentations are "of pullback type" or arise via middle convolution fromfinite complex reflection groups. In particular, we classify all rank 2 localsystems of geometric origin on the projective line with n generic punctures,and with local monodromy of infinite order about at least one puncture.

Finite braid group orbits on $SL_2$-character varieties