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