  Using the formalism of supersymmetric quantum mechanics, we obtain a large
number of new analytically solvable one-dimensional periodic potentials and
study their properties. More specifically, the supersymmetric partners of the
Lame potentials ma(a+1)sn^2(x,m) are computed for integer values a=1,2,3,....
For all cases (except a=1), we show that the partner potential is distinctly
different from the original Lame potential, even though they both have the same
energy band structure. We also derive and discuss the energy band edges of the
associated Lame potentials pm sn^2(x,m)+qm cn^2(x,m)/ dn^2(x,m), which
constitute a much richer class of periodic problems. Computation of their
supersymmetric partners yields many additional new solvable and quasi exactly
solvable periodic potentials.
