Spectral Curve of the Halphen Operator

The Halphen operator is a third-order operator of the form $L_3=\partial_x^3-g(g+2)\wp(x)\partial_x-\frac{1}{2}g(g+2)\wp'(x),$ where $g\ne 2\,\mbox{mod(3)}$, the Weierstrass $\wp$-function satisfies the equation $(\wp'(x))^2=4\wp^3(x)-g_2\wp(x)-g_3.$ In the equianharmonic case, i.e., $g_2=0$ the Halphen operator commutes with some ordinary differential operator $L_n$ of order $n\ne 0\,\mbox{mod(3)}.$ In this paper we find the spectral curve of the pair $L_3,L_n$.

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