On the Spectral Curve of the Halphen Operator    [PDF]

Andrey E. Mironov, Dafeng Zuo
The Halphen operator is a third-order operator introduced in \cite{H} and of the form \beq L_3=\dfrac{d^3}{dx^3}-g(g+2)\wp(x)\dfrac{d}{dx}-\frac{g(g+2)}{2}\wp'(x),\nn \eeq where $g\ne 2\,\mbox{mod(3)}$ and $\wp(x)$ is the Weierstrass $\wp$-function satisfying the equation $(\wp'(x))^2=4\wp^3(x)-g_2\wp(x)-g_3.$ In this paper, we find a recursive formula for the equation of the spectral curve of the Halphen operator in the equinaharmonic case, $i.e.$, $g_2=0$.
View original: http://arxiv.org/abs/1305.6267