Lame equations with algebraic solutions

In this paper we study Lame equations L-n,L-By = 0 in so-called algebraic form, having only algebraic functions as solution. In particular we provide a complete list of all finite groups that occur as the monodromy groups, together with a list of examples of such equations. We show that the set of such Lame equations with n is not an element of 1/2 + Z is countable, up to scaling of the equation. This result follows from the general statement that the set of equivalent second-order equations, having algebraic solutions and all of whose integer local exponent differences are 1, is countable. (C) 2003 Elsevier Inc. All rights reserved.