  We describe globally nilpotent differential operators of rank 2 defined over
a number field whose monodromy group is a nonarithmetic Fuchsian group. We show
that these differential operators have an S-integral solution. These
differential operators are naturally associated with Teichmueller curves in
genus 2. They are counterexamples to conjectures by Chudnovsky--Chudnovsky and
Dwork. We also determine the field of moduli of primitive Teichmueller curves
in genus 2, and an explicit equation in some cases.
