Monodromy of a generalized Lame equation of third order

Abstract: We study the monodromy of the following third order linear differential equation [y'''(z)-(\alpha\wp(z;\tau)+B)y'(z)+\beta\wp'(z;\tau)y(z)=0, ] where $B\in\mathbb{C}$ is a parameter, $\wp(z;\tau)$ is the Weierstrass $\wp$-function with periods $1$ and $\tau$, and $\alpha,\beta$ are constants such that the local exponents at the singularity $0$ are three distinct integers, which can always be written as $-n-l, 1-l, n+2l+2$ after a dual transformation, where $n,l\in\mathbb{N}$. This ODE can be seen as the third order version of the well-known Lam\'{e} equation $y''(z)-(m(m+1)\wp(z;\tau)+B)y(z)=0$. We say that the monodromy is unitary if the monodromy group is conjugate to a subgroup of the unitary group. We show that \begin{itemize} \item[(i)] if $n, l$ are both odd, then the monodromy can not be unitary; \item[(ii)] if $n$ is odd and $l$ is even, then there exist finite values of $B$ such that the monodromy is the Klein four-group and hence unitary; \item[(iii)] if $n$ is even, then whether there exists $B$ such that the monodromy is unitary depends on the choice of the period $\tau$. \end{itemize} The methods of studying the second order Lam\'{e} equation can not work here, and we need to develop different approaches to treat these different cases separately. These results have interesting applications to the integrable $SU(3)$ Toda system in another work (Chen-Lin, J. Differ. Geom. to appear).