Heun operator of Lie type and the modified algebraic Bethe ansatz

The generic Heun operator of Lie type is identified as a certain $BC$-Gaudin magnet Hamiltonian in magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain spectrum terms roots inhomogeneous equations. We show also that these are intimately associated polynomial solutions differential equation. illustrate use this approach two contexts: representation theory $O(3)$ and computation entanglement entropy for free Fermions on Krawtchouk chain.