  We propose a new approach based on the algebraization of the Associated
Lam\'{e} equation \[-\psi''(x) + [ m(m+1)k^{2}sn^{2}x +
\ell(\ell+1)k^{2}(cn^{2}x/dn^{2}x)]\psi(x) = E\psi(x)\] within sl(2,R) to
derive the corresponding periodic potentials. The band edge eigenfunctions and
energy spectra are explicitly obtained for integers m,$\ell$. We also obtain
the explicit expressions of the solutions for half-integer m and integer or
half-integer $\ell$.
