  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.
