  We give a complete characterization of the classical Lam\'e equations $y'' =
(n(n + 1)\wp(z) + B)y$, $n \in \Bbb R$, $B \in \Bbb C$ on flat tori $E_\tau =
\Bbb C/(\Bbb Z + \Bbb Z\,\tau)$ with finite monodromy groups $M$.
  Beuker--Waall had shown that such $n$ must lie in a finite number of
arithmetic progressions $n_i + \Bbb N \subset \Bbb Q$ and they determined all
corresponding $M$. By combining the theory of dessin d'enfants with the
geometry of spherical tori, we prove the existence of $(B, \tau)$ for each such
$n$ and provide a description of all such $(n, B, \tau, M)$.
  In particular, for a given $(n, M)$ with $n \not\in \tfrac{1}{2} + \Bbb Z$,
we prove the finiteness of $(B, \tau)$ and derive an explicit counting formula
of them. (The case $n \in \tfrac{1}{2} + \Bbb Z$ is a classical result due to
Brioschi--Halphen--Crawford.)
  The main ingredients in this work are (1) the definition and classification
of basic spherical triangles with finite monodromy and (2) the process of
attaching cells corresponding to $n \mapsto n + 1$ which reduces the problem to
the basic case.
