Lamé equations with algebraic solutions

Abstract

In this paper we study Lamé equations L_n,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 Lamé equations with $\text{n∉1/2+}\text{Z}$ 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.