Moduli of Higgs bundles over the five punctured sphere

We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector $\mu\in[0,1]^5$. The moduli space corresponding to the central weight $\mu_c=(\frac{1}{2}, \dots, \frac{1}{2})$ is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the $\mathbb C^*$-action we obtain a proof of Simpson's foliation conjecture in this case. For each $n\ge 5$, we remark that there is a weight vector so that the foliation conjecture in the moduli space of rank two logarithmic connections over the projective line minus $n$ points is false.