Halphen's transform and middle convolution

We show that the Halphen transform of a Lam'e equation can be written as the symmetric square of the Lam'e equation followed by an Euler transform. We use this to compute a list of Lam'e equations with arithmetic Fuchsian monodromy group. It contains all those Lam'e equations where the quaternion algebra $A$ over $k$ associated to the arithmetic Fuchsian group is a quaternion algebra $A$ over $\mathbb{Q}$. Further we classify all geometric braid group orbits in ${\mathrm{S}\mathrm{L}}_2(\text{ZZ}{)}^4$ with the possible exception of three orbits.