Rational Calogero–Moser critical-point conjecture for cyclic operators

Let $\ell\ge 2$ and let $L=\partial^\ell+a_2(z)\partial^{\ell-2}+\dots+a_\ell(z)$ be a rational differential operator invariant under $\mathbb Z_\ell$, with coefficients vanishing at infinity. Suppose its nonzero poles are representatives $z_1,\dots,z_N$ of distinct $\mathbb Z_\ell$-orbits and their images, with arbitrary integer indices at $0$ and indices $-1,1,\dots,\ell-2,\ell$ at the nonzero poles. Rational Calogero–Moser conjecture. Algebraically integrable operators with these properties correspond to critical points of the degree-$\ell$ rational Calogero–Moser Hamiltonian for the complex reflection group $S_N\ltimes\mathbb Z_\ell^N$. The source notes that this conjecture holds for $\ell\le 3$; its general status remains open.