  In this paper we present an algebraic study concerning the general second
order linear differential equation with polynomial coefficients. By means of
Kovacic's algorithm and asymptotic iteration method we find a degree
independent algebraic description of the spectral set: the subset, in the
parameter space, of Liouiville integrable differential equations. For each
fixed degree, we prove that the spectral set is a countable union of non
accumulating algebraic varieties. This algebraic description of the spectral
set allow us to bound the number of eigenvalues for algebraically
quasi-solvable potentials in the Schr\"odinger equation.
