  We show that the existence of algebraic forms of quantum, exactly-solvable,
completely-integrable $A-B-C-D$ and $G_2, F_4, E_{6,7,8}$ Olshanetsky-Perelomov
Hamiltonians allow to develop the {\it algebraic} perturbation theory, where
corrections are computed by pure linear algebra means. A Lie-algebraic
classification of such perturbations is given. In particular, this scheme
admits an explicit study of anharmonic many-body problems. The approach also
allows to calculate the ratios of a certain generalized Dyson-Mehta integrals
algebraically, which are interested by themselves.
