This chapter introduces an alternative framework for solving spectral problems in classical and quantum integrable systems via separation of variables, grounded in the Hamilton–Jacobi theory. The method is first outlined in the classical context and then extended to quantum integrable spin chains with glell symmetry and associated Yangians.

A key objective is to construct separated representations for the spectrum, reducing multidimensional eigenvalue problems to a set of one-dimensional ones solvable through algebraic and representation-theoretic techniques. This approach proves especially effective in cases where the completeness of the Bethe ansatz is unclear, offering complementary insights.

Focusing on the classical glell magnets, the chapter details the construction of separated variables using the Baker–Akhiezer function and its role in identifying canonical coordinates on the spectral curve. In the quantum case, complete diagonalization of the gl2 spin chain is achieved through separation of variables.

For higher-rank models and generic local representations of glell, the full construction remains an open and active area of research. The chapter closes with an overview of current developments and challenges in advancing this method.