Algebraic Solutions of the Lamé Equation, Revisited

A minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, “On algebraic solutions of Lamé’s differential equation”, J. Differential Equations 41 (1) (1981), 44–58.] It is shown that if the group is the octahedral group , then the degree parameter of the equation may differ by from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation. [See R. C. Churchill, “Two-generator subgroups of and the hypergeometric, Riemann, and Lamé equations”, J. Symbolic Computation 28 (4–5) (1999), 521–545.] The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.