Lame operators with projective octahedral and icosahedral monodromies

We show that there exists a Lame operator $L_n$ with projective octahedral monodromy for each $n\in{1/2}(\mathbf{N}+{1/2})\cup{1/3}(\mathbf{N}+{1/2})$, and with projective icosahedral monodromy for each $n\in{1/3}(\mathbf{N}+{1/2})\cup{1/5}(\mathbf{N}+{1/2})$. To this end, we construct Grothendieck's dessin d'enfants corresponding to the Belyi morphisms which pull-back hypergeometric operators into Lame operators $L_n$ with the desired monodromies.