Algebraic-rational nature of two-dimensional, Weyl-invariant integrable quantum Hamiltonians (rational, trigonometric and elliptic) is revealed and reviewed. If written in special Weyl invariants (polynomial, exponential and elliptic), all similarity-transformed Hamiltonians (and integrals) are in algebraic form: they are the second order differential operators with polynomial coefficients; the flat metric in the Laplace-Beltrami operator is polynomial, their potentials are rational functions with singularities at the boundaries of the configuration space. Ground state eigenfunctions are algebraic functions in a form of product of polynomials in some degrees. par It is shown that $A_2$ and $BC_2$ models possess the hidden algebra $gl(3)$, their coupling constants are defined by spin of representation. For quantized values of coupling constants the representation becomes finite-dimensional and a number of polynomial eigenfunctions occur. par For the elliptic Hamiltonian there exists a single finite-dimensional invariant subspace. While the rational and trigonometric Hamiltonians preserve the same infinite flag of polynomial spaces. Unusual particular integral common for all three models is derived.