Quasi-exact-solvability of the $A_{2}/G_2$ Elliptic model: algebraic forms, $sl(3)/g^{(2)}$ hidden algebra, polynomial eigenfunctions

The potential of the $A_2$ quantum elliptic model (3-body Calogero-Moser elliptic model) is defined by the pairwise three-body interaction through Weierstrass $\wp$-function and has a single coupling constant. A change of variables has been found, which are $A_2$ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown that the model possesses the hidden $sl(3)$ algebra - the Hamiltonian is an element of the universal enveloping algebra $U_{sl(3)}$ for arbitrary coupling constant - thus, it is equivalent to $sl(3)$-quantum Euler-Arnold top. The integral, in a form of the third order differential operator with polynomial, is constructed explicitly, being also an element of $U_{sl(3)}$. It is shown that there exists a discrete sequence of the coupling constants for which a finite number of polynomial eigenfunctions, up to a (non-singular) gauge factor occur. The potential of the $G_2$ quantum elliptic model (3-body Wolfes elliptic model) is defined by the pairwise and three-body interactions through Weierstrass $\wp$-function and has two coupling constants. A change of variables has been found, which are $G_2$ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown the model possesses the hidden $g^{(2)}$ algebra. It is shown that there exists a discrete family of the coupling constants for which a finite number of polynomial eigenfunctions up to a (non-singular) gauge factor occur.