Abstract:  Reviewed the research history and main methods of quasi exactly solvable systems The QES system falls between precisely solvable and unsolvable systems and is capable of solving some eigenvalues and eigenstates under specific conditions The research began with the Razavy potential and gradually developed through Lie algebra, Heun equations, supersymmetric quantum mechanics, and the Bethe Ansatz method These analytical tools reveal the symmetry and complex structure of QES systems, expanding the scope of solvable models The article also discusses the application of pre potential methods and possible future research directions for QES systems, including non Hermitian quantum systems and experimental verification The research on QES system has a profound impact on multiple fields of physics

Key words: quasi exactly solvable, heun equation, bethe ansatz method