QUASI EXACTLY SOLVABLE EXTENSION OF THE LAME EQUATION

Quasi exactly solvable equations are spectral equations which possess a finite number of algebraic eigenvectors. For a few of these equation s, such as the Lame equation, the number of algebraic solutions is lar ger than predicted by the general theory. A class of quasi exactly sol vable equations in one variable is considered and the conditions under which a rich algebraization occurs is discussed. Families of equation s, which by themselves are not quasi exactly solvable, but can be tran sformed into systems of coupled quasi exactly solvable equations, are discussed. These results suggest a scheme for the classification of qu asi exactly solvable systems in one variable.