  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
$\rm{SL}_2(\ZZ)^4$ with the possible exception of three orbits.
