Algebraic quantum Hamiltonians on the plane. Polynomial form for the elliptic $A_n$ Calogero-Moser system

Algebraic quantum Hamiltonians on the plane
We consider second order differential operators $P$ with polynomial coefficients that preserve the vector space $V_k$ of polynomials of degrees not greater then $k$. Additionally we assume that the metric associated with the symbol of $P$ is flat and that the operator $P$ is potential. In the case of two independent variables we obtain some classification results and find polynomial forms for the elliptic $A_2$ and $G_2$ Calogero-Moser Hamiltonians and for the elliptic Inozemtsev model.
Polynomial form for the elliptic $A_n$ Calogero-Moser system.
A conjecture on a change of variables that brings the $A_n$ elliptic Calogero-Moser Hamiltonian to a polynomial form is formulated. In the case $n=1,2,3$ a proof is presented.